generalized whittaker functions on gsp ;r) …moriyama/moriyama-jmsj-2011.pdfc 2011 the mathematical...

c©2011 The Mathematical Society of JapanJ. Math. Soc. JapanVol. 63, No. 4 (2011) pp. 1203–1262doi: 10.2969/jmsj/06341203

Generalized Whittaker functions on GSp(2,R) associated

with indefinite quadratic forms

By Tomonori Moriyama

(Received July 19, 2009)(Revised May 19, 2010)

Abstract. We study the generalized Whittaker models for G =GSp(2,R) associated with indefinite binary quadratic forms when they arisefrom two standard representations of G: (i) a generalized principal series rep-resentation induced from the non-Siegel maximal parabolic subgroup and (ii) a(limit of) large discrete series representation. We prove the uniqueness of suchmodels with moderate growth property. Moreover we express the values of thecorresponding generalized Whittaker functions on a one-parameter subgroupof G in terms of the Meijer G-functions.

0. Introduction.

Let G = GSp(2) be the symplectic group with similitude defined over thefield Q of rational numbers. When we wish to write down the Fourier expan-sion of automorphic forms on GA along its Siegel parabolic subgroup, we neces-sitate not only the Whittaker models but also the generalized Whittaker mod-els (see Section 1 for details). Our concern in this paper is the local theory ofgeneralized Whittaker models at the real place, which still lies in an intermedi-ate state. For example, in the paper [PS], which had circulated since the late1970s, I. I. Piatetski-Shapiro stated the multiplicity free theorem of such modelsfor G := GR = GSp(2,R) without a proof ([PS, Theorem 3.1]). However nobodyseems to establish it up to today. For the generalized Whittaker model for G

associated with a definite binary quadratic form, there are some results support-ing Piatetski-Shapiro’s assertion. Besides H. Yamashita’s result [Y] in a generalsetting, there are several detailed studies for specific kinds of representations of G

([Ni], [Mi-1], [Mi-2], [Is]), where the multiplicity free results as well as explicitformulae of generalized Whittaker functions are obtained (see Subsection 8.3). Onthe other hand, little is known about the generalized Whittaker models associated

2000 Mathematics Subject Classification. Primary 11F46; Secondary 22E30.

Key Words and Phrases. generalized Whittaker functions, Fourier expansions of automor-phic forms on GSp(2), the Meijer G-functions.

This research was partially supported by Grant-in-Aid for Scientific Research (No. 17740023and 20740015), Japan Society for the Promotion of Science.

http://dx.doi.org/10.2969/jmsj/06341203

1204 T. Moriyama

with indefinite binary quadratic forms, although they are equally important in thestudy of automorphic forms on GA. The purpose of this paper is to study themfor two kinds of standard representations of G.

To be more precise, let P = M nN be the (standard) Siegel parabolic sub-group of G. For each symmetric matrix β = tβ ∈ Mat(2)R, we define a characterψβ : NR → C(1) by

ψβ

((I2 x

02 I2

))= exp

(2π√−1 tr(βx)

).

Suppose that the character ψβ is non-degenerate (i.e. det(β) 6= 0). We denoteby Mβ the identity component (as an algebraic group) of the stabilizer of ψβ inM and set Rβ := Mβ,R nNR. For each quasi-character χ of Mβ,R, we have aquasi-character χ · ψβ of Rβ and an induced representation

C∞(Rβ\G;χ · ψβ)

:=W : G

C∞−→ C | W (rg) = (χ · ψβ)(r)W (g), ∀(r, g) ∈ Rβ ×G,

on which G acts by right translation. The totality of functions in C∞(Rβ\G;χ·ψβ)having moderate growth property is denoted by C∞mg(Rβ\G;χ · ψβ). We denotethe Lie algebra of G by g and take a (standard) maximal compact subgroup K ofG. For a quasi-simple (g,K)-module (π, Hπ), we set

GWG(π, χ · ψβ) := Homg,K(Hπ, C∞(Rβ\G;χ · ψβ)),

which is called the space of generalized Whittaker functionals for π. By a general-ized Whittaker function belonging to π, we understand the image Φ(v) of a vectorv ∈ Hπ under some Φ ∈ GWG(π, χ · ψβ). If Φ ∈ GWG(π, χ · ψβ) belongs to thesubspace

GWmgG (π, χ · ψβ) := Homg,K

(Hπ, C∞mg(Rβ\G;χ · ψβ)

),

then we say Φ and Φ(v) have the moderate growth property. From the view pointof automorphic forms, the following two problems are fundamental:

Problem (A). Estimate the dimensions of GWG(π, χ · ψβ) andGWmg

G (π, χ · ψβ). In particular, determine whether the multiplicity free prop-erty dimC GWmg

G (π, χ · ψβ) ≤ 1 holds or not.

Generalized Whittaker functions 1205

Problem (B). Find tractable formulae of the generalized Whittaker func-tions Φ(v) for Φ ∈ GWmg

G (π, χ · ψβ) and appropriate vectors v ∈ Hπ.

In connection with Problem (B), the following problem is also important.

Problem (C). Construct a non-zero element in GWmgG (π, χ·ψβ), especially

when the estimate dimC GWmgG (π, χ · ψβ) ≤ 1 holds.

Let us summarize the current status of these problems according as the sig-nature of det(β).

The case of det(β) > 0. As we mentioned above, there are some posi-tive results on the Problem (A) in this case. In the first place, H. Yamashita[Y] obtained a result closely related to the Problem (A), which we now explain.For an irreducible unitary representation (π, Hπ) of G, we consider the space ofcontinuous intertwining operators

GW∞G (π, χ · ψβ) := HomG(H ∞

π , C∞(Rβ\G;χ · ψβ)

).

Here H ∞π stands for the smooth vectors in Hπ equipped with the usual C∞-

topology. Note that there is a natural inclusion GW∞G (π, χ · ψβ) ⊂ GWmg

G (π, χ ·ψβ) (cf. [Wal, Section 5.1]). Then a result of H. Yamashita ([Y, Theorem 6.9 (3)]),who works for general connected semisimple groups of hermitian type, implies thatdimC GW∞

G (π, χ · ψβ) ≤ 1 under the following three conditions: det(β) > 0, therestriction of π to G0 := Sp(2,R) remains irreducible, and the restriction of χ

to Mβ,R ∩ G0(∼= O(2)) is real-valued. Meanwhile, in the early 1990s, S. Niwa[Ni] constructed the generalized Whittaker functions on G associated with defi-nite quadratic forms by using theta liftings. He gives an integral expression of thegeneralized Whittaker function on G belonging to the spherical principal series rep-resentation. T. Miyazaki [Mi-1], [Mi-2] and T. Ishii [Is] extended Niwa’s study toseveral standard representations and some derived functor modules by construct-ing differential equations satisfied by the generalized Whittaker functions. Morerecently, D. Prasad and R. Takloo-Bighash [Pr-TB], [TB] determine the dimen-sion of GW∞

G (π, χ · ψβ) for the discrete series representations π when det(β) > 0by using theta liftings. In a paper of A. Pitale and R. Schmidt [Pi-Sch], the caseof holomorphic discrete series representations is also studied.

The case of det(β) < 0. First we note that GWmgG (π, χ · ψβ) = 0 for

a holomorphic discrete series representation π of G (cf. [Pi-Sch]), which can beseen as a paraphrase of Koecher’s principle for holomorphic automorphic forms.In [Mi-2], the case of some derived functor modules with small Gel’fand-Kirillovdimensions are studied. However there seems to be no detailed investigation for

1206 T. Moriyama

standard representations. In fact, a recent investigation with T. Ishii tells usthat Problem (A) becomes more subtle when det(β) < 0. To explain it, supposethat π is an irreducible principal series representation of G induced from the Borelsubgroup. Then it seems possible to construct two linearly independent functionalsin

Homg0,K0


), g0 := sp(2,R), K0 := G0 ∩O(4)

by using the Novodvorsky local zeta integrals (cf. Section 9). Nevertheless, we canstill prove that dimC GWmg

G (π, χ · ψβ) ≤ 1 thanks to the non-trivial outer auto-morphism of Sp(2,R). This phenomenon, which we do not encounter whendet(β) > 0, suggest that the appropriate group for multiplicity free theorem shouldbe the disconnected group G = GSp(2,R), not the connected group Sp(2,R).

Now we are in the position to state our main results of this paper:

Main Results (see Theorems 5.1, 6.1, 7.1, and 8.1 for precise statements).Suppose that det(β) < 0 and take an arbitrary quasi-character χ of Mβ,R. Let(π, Hπ) be either (i) an irreducible generalized principal series representation ofG induced from the non-Siegel maximal parabolic subgroup P1 (an irreducibleP1-principal series representation) or (ii) a (limit of) large discrete series represen-tation. Then we have the following assertions:

(1) The space GWmgG (π, χ · ψβ) is at most one dimensional.

(2) If π is equivalent to an irreducible P1-principal series representation, then wehave dimC GWG(π, χ · ψβ) ≤ 4.

(3) For an element Φ ∈ GWmgG (π, χ · ψβ), the values of the generalized Whit-

taker function Φ(v) corresponding to some specific vector v ∈ Hπ on a one-parameter subgroup of G can be expressed by the Meijer G-function G4,0

2,4(z).

The main results should play the following important roles in automorphicforms. In the first place, the multiplicity free theorem (1) allows us to express theglobal generalized Whittaker function arising from a Hecke eigen form on GA asa product of the generalized Whittaker function on GR and a function on GAf .Besides, there are several automorphic L-functions whose integral representationsinvolve generalized Whittaker functions on GA (e.g. [An], [PS], [An-Ka],[PS-Ra], [F]). In fact, the direct motivation of our investigation is to extendour earlier results [Mo] on the spinor L-function to a cusp form not having globalWhittaker models by evaluating the real local zeta integrals of Andrianov ([An],[PS]). Note that our explicit formulae in (3) are quite suitable for this purpose,because they are given by inverse Mellin transforms, the Meijer G-functions. Wehope to discuss this issue in a future paper, which is also an indispensable step to


study arithmetic properties of the spinor L-function (e.g. [Ha], [Le]).Our proof of the main results is done by analyzing the differential equations

satisfied by the generalized Whittaker functions, which is basically parallel to[Mi-1], [Mi-2], and [Is]. Besides, there are two crucial points in our proof. One isthe fact that the generalized Whittaker function with moderate growth propertydecreases rapidly in a certain direction (Lemma 3.3). The other is the determina-tion of rapidly decreasing solutions of a fourth order generalized hypergeometricequation (Proposition 9.2). Thanks to Proposition 9.2, we can prove our mainresults without assuming any conditions on χ. This is quite satisfactory when weapply our results to automorphic forms.

The organization of this paper is as follows. In Section 1, we introduce aFourier expansion of automorphic forms on GA involving the usual and general-ized Whittaker functions, which motivates us to study the generalized Whittakermodels. In particular, we prove the absolute convergence of the integral definingthe global generalized Whittaker function arising from a cusp form, which is adelicate problem when −det(β) ∈ (Q×)2. Moreover we give several equivalentconditions for a cusp form on GA to have a non-zero global Whittaker function(Proposition 1.2). In Section 2 we collect some basic notation concerning the Liegroup G0 and introduce two kinds of standard representations of G in the mainresults. In Section 3, we define the generalized Whittaker functions on G0. We dis-cuss the restriction of generalized Whittaker functions to a two-dimensional splittorus S of G0 satisfying G0 = (Rβ∩G0)SK0 for det(β) < 0. We also prove the keyobservation (Lemma 3.3) mentioned above. Our proof of the main results occupiesSection 4–Section 8. In Section 4, we introduce two kinds of differential operatorsacting on the generalized Whittaker functions, the shift operators and the Casimiroperators. We compute the S-radial part of these two differential operators. InSection 5 and Section 6 (resp. Section 7) we treat P1-principal series representa-tions (resp. (limits of) large discrete series representations). In each section, wederive a system of differential equations for the generalized Whittaker function byusing differential operators constructed in Section 4. By a careful analysis of thesedifferential equations, we obtain our explicit formulae. In Section 8, after clarify-ing the relation between the generalized Whittaker functions on G0 and those onG, we prove the multiplicity free theorem (Theorem 8.2) for det(β) < 0. We alsoprove the multiplicity free theorem for det(β) > 0 (Theorem 8.3), which is a re-formulation of the results of T. Miyazaki [Mi-1]. In Section 9, we state and provethe above-mentioned key result (Proposition 9.2) on generalized hypergeometricdifferential equations. In the final section, we propose an approach to Problem(C) via Novodvorsky’s local zeta integrals ([No]) when det(β) < 0.

Notation and Conventions.

(i) For each place v of the field Q of rational numbers, we denote by Qv the

1208 T. Moriyama

completion of Q at v. The module of an element x ∈ Qv is denoted by |x|v.For a finite place p of Q, Zp stands for the ring of integers in Qp. The adelering (resp. the ring of finite adeles, the idele group) is denoted by A (resp. Af ,A×). The module of an element x ∈ A× is denoted by |x|A or simply by |x|.Unless otherwise stated, we understand that all the measures on locally compactunimodular groups are the Haar measures.

(ii) Let G = GSp(2) be the symplectic group with similitude of degree two,which is defined by

G :=g ∈ GL(4) | tgJ4g = ν(g)J4 for some ν(g) ∈ Gm

, J4 :=

(02 I2

−I2 02

).

We regard G as an algebraic group defined over Q. For any Q-algebra R, thegroup of R-valued points of G is denoted by GR. We adopt the same conventionfor other algebraic groups. For each element g = (gv) = (gv,i,j)1≤i,j≤4 ∈ GA, wedefine its norm ‖g‖ by

‖g‖ :=∏v

‖gv‖v with ‖gv‖v := max|gv,i,j |v, |(g−1

v )i,j |v | 1 ≤ i, j ≤ 4.

As a maximal compact subgroup KA of GA, we take

KA :=∏v

Kv with K∞ := O(4) ∩GR and Kp := GL(4,Zp) ∩GQp (∀p < ∞).

We also use the following notation:

G := GR = GSp(2,R), G0 ≡ Sp(2,R) = g ∈ G | ν(g) = 1,K = K∞, K0 := K ∩G0.

We write the Lie algebras of G, G0, and K0 by g, g0, and k, respectively. If thereis no fear of confusion, we do not distinguish a smooth representation of G (resp.G0) from its underlying (g,K)-module (resp. (g0,K0)-module). For an arbitraryLie subalgebra l of g, its complexification l⊗C is denoted by lC . We denote thedual space HomC(lC ,C) of lC by l∗C .

(iii) Let L be a Lie group with Lie algebra l = Lie(L). For a C∞-function f

on L, we set

[RXf ](x) :=d

dt

∣∣∣∣t=0

f(x exp(tX)), X ∈ l, x ∈ L.


This action of l can be extended to that of the universal enveloping algebra U(l)of l. We also write f(x;X) for [RXf ](x) (X ∈ U(l)).

(iv) For a positive integer n, Mat(n) stands for the algebraic group of n × n

matrices defined over Q.

Acknowledgments. I would like to express my gratitude to ProfessorTakayuki Oda, who explained me importance of generalized Whittaker modelsmore than ten years ago. I am grateful for Professors Masaaki Furusawa, TakuIshii, and Masao Tsuzuki for their interests and fruitful discussions on this work.Thanks also go to Dr. Tadashi Miyazaki, who informed me of an error in theearlier version of this paper. I wish to thank the referees for suggesting someimprovements in the original version of this paper.

1. Fourier expansions of automorphic forms on GSp(2).

In this section, we formulate a Fourier expansion of automorphic forms onGSp(2)A along the Siegel parabolic subgroup (cf. [PS], [Su]) in order to motivateour study of generalized Whittaker functions on GSp(2,R). The expansions areexpressed in terms of the global Whittaker functions and the global generalizedWhittaker functions. Although almost all materials here might be found in theliterature, we put them together for the sake of convenience. Moreover we giveinteresting equivalent conditions for a cusp form to have a non-zero global Whit-taker function (Proposition 1.2).

1.1. The first step of the Fourier expansion.The center of G is given by Z := z14 ∈ G | z ∈ Gm. We denote the space of

automorphic forms on GA (resp. the space of cusp forms on GA) with central char-acter ω : Q×\A× → C(1) by A (GQZA\GA;ω) (resp. A cusp(GQZA\GA;ω)).We fix a maximal parabolic subgroup P of G corresponding to the short root(the so-called Siegel parabolic subgroup) and its Levi decomposition P = MN asfollows:

P :=( ∗ ∗

02 ∗)∈ G

,

M :=

m(h, λ) :=(

h 02

02 λth−1

)∣∣∣∣ h ∈ GL(2), λ ∈ Gm

,

N :=(

I2 x

02 I2

)∣∣∣∣ x ∈ Sym(2)

,

where we set Sym(2) := x ∈ Mat(2) | tx = x. Let ψ : Q\A → C(1) be a

1210 T. Moriyama

non-trivial character of Q\A characterized by ψ(t∞) = exp(2π√−1t∞) (t∞ ∈ R).

For each β ∈ Sym(2)Q, we define a character ψβ of NA by

ψβ

((I2 x

02 I2

))= ψ(tr(βx)).

An automorphic form F ∈ A (GQZA\GA;ω) have the following Fourier expansion

F (g) =∑

β∈Sym(2)Q

Fβ(g), Fβ(g) :=∫

NQ\NA

dnF (ng)ψβ(n)−1. (1.1)

For each β ∈ Sym(2)Q, the β-th coefficient function Fβ(g) satisfies the relation

Fβ(ng) = ψβ(n)Fβ(g), ∀(n, g) ∈ NA ×GA.

In order to describe relations between the coefficient functions Fβ |β ∈ Sym(2)Q,we introduce the right action of M on Sym(2) by

β ·m(h, λ) := λ−1thβh, (β ∈ Sym(2), m(h, λ) ∈ M). (1.2)

Lemma 1.1. Take two symmetric matrices β, β′ ∈ Sym(2)Q. Suppose thatthere exists an element m ∈ MQ such that β′ = β ·m. Then we have

Fβ′(g) = Fβ(mg), ∀g ∈ GA.

For β, β′ ∈ Sym(2)Q, we write β ∼MQβ′ if there exists an element

m ∈ MQ such that β′ = β · m. In view of Lemma 1.1, an automorphicform F ∈ A (ZAGQ\GA;ω) is determined by the coefficient functions Fβ(g) |β ∈ Sym(2)Q/ ∼MQ

. A complete set of representatives for the coset spaceSym(2)Q/ ∼MQ

is given by

02 ∪

β(1) :=(

0 00 1

)∪

β

(2)d :=

(1 00 −d

) ∣∣∣∣ d ∈ Q×/(Q×)2

. (1.3)

Suppose that F is a cusp form. Then the function F02(g) vanishes identically,because it is nothing but the constant term of F along P. Hence the equation(1.1) tells us that at least one of the following three conditions holds:

(Rk1): there exists a non-zero symmetric matrix β ∈ Sym(2)Q such that det(β) =


0 and Fβ(g) 6≡ 0;(Rk2-D): there exists a symmetric matrix β ∈ Sym(2)Q such that det(β) > 0 andFβ(g) 6≡ 0;(Rk2-ID): there exists a symmetric matrix β ∈ Sym(2)Q such that det(β) < 0 andFβ(g) 6≡ 0.

Remark. A result of J. S. Li [Li] tells us that an arbitrary non-zero cuspform F ∈ A cusp(GQZA\GA;ω) satisfies either (Rk2-D) or (Rk2-ID). This can beseen from Proposition 1.2 below, too. It is well known that a holomorphic cuspform F on GA satisfies neither (Rk1) nor (Rk2-ID).

In the next two subsections, we shall express the functions Fβ(g) (β 6= 02)in terms of the global Whittaker functions and the global generalized Whittakerfunctions to get a finer expansion of F (g).

1.2. The second step of the Fourier expansion, the case of det(β)= 0.

In this subsection, we consider the case where β ∈ Sym(2)Q satisfies det(β) =0 and β 6= 02. By Lemma 1.1, we may assume that β = β(1) =

(0 00 1

). We fix a

maximal unipotent subgroup of G defined over Q as follows:

N0 :=

n(x0, x1, x2, x3) :=

1 x1 x2

1 x2 x3

11

1 x0

11−x0 1

∈ G

.

For each g ∈ GA, we put hF (x0; g) = Fβ(1)(n(x0, 0, 0, 0)g), (x0 ∈ A/Q). Applyingthe Fourier inversion formula to hF (x0; g), we have

Fβ(g) =∫

Q\AhF (x0; g)dx0 +

∑

α∈Q×WF (diag(α, 1, α−1, 1)g). (1.4)

Here WF (g) is the global Whittaker function attached to F defined by

WF (g) :=∫

Q\AhF (x0; g)ψ(x0)−1dx0

=∫

(Q\A)4F (n(x0, x1, x2, x3)g)ψ(x0 + x3)−1dx1dx2dx3dx0. (1.5)

An automorphic form F on GA is said to be globally generic if the global Whit-

1212 T. Moriyama

taker function WF (g) attached to F does not vanish identically. We give severalcharacterizations of globally generic cusp forms:

Proposition 1.2. For a cusp form F (g) on GA, the following four condi-tions are equivalent :

( i ) F is globally generic;( ii ) the condition (Rk1) holds;(iii) for each β ∈ Sym(2)Q satisfying −det(β) ∈ (Q×)2, we have Fβ(g) 6≡ 0;(iv) the integral FZ(g) :=

∫Q\A F (n(0, x1, 0, 0)g)dx1 (g ∈ GA) does not vanish

identically.

Proof. (i) ⇒ (ii): This is immediate from the definition (1.5) of the globalWhittaker function. (ii)⇒ (i): Since F is assumed to be a cusp form, the first termof the right hand side of (1.4) vanishes. This implies that the global Whittakerfunction WF (g) does not vanish identically. (i) ⇒ (iii): Define two elements w1

and w2 of GQ by

w1 :=

11

11

and w2 :=

1−1

11

. (1.6)

Then we have

WF (g)

=∫

(Q\A)4F

(n(x0, x1, 0, 0)n(0, 0, x2, x3)g

)ψ(−x0 − x3)dx1dx0dx2dx3

=∫

(Q\A)4F

(w−1

2 n(x0, x1, 0, 0)n(0, 0, x2, x3)g)ψ(−x0 − x3)dx1dx0dx2dx3

=∫

(Q\A)4F

(n(0, x1,−x0, 0)w−1

2 n(0, 0, x2, x3)g)ψ(−x0 − x3)dx1dx0dx2dx3.

Therefore we know that

∫

(Q\A)2F

(n(0, x1, x0, 0)g

)ψ(x0) dx0dx1 6≡ 0.

Hence it follows from the equality


∫

(Q\A)2F

(n(0, x1, x2, 0)g

)ψ(x2) dx1dx2

=∑

a∈Q

∫

(Q\A)3F

(n(0, x1, x2, x3)g

)ψ(x2 + ax3) dx1dx2dx3

that there exists a ∈ Q such that Fβ(g) 6= 0 for β =( 0 −1/2−1/2 −a

). By Lemma 1.1,

this implies (iii). Finally the implications (iii) ⇒ (iv) ⇒ (i) are proved in [K-R-S,Lemma 8.2], for example. ¤

Remark. The above proof shows that the implications (i) ⇒ (ii) and (i) ⇒(iii) hold even if F is not a cusp form. The converse implications (ii) ⇒ (i) and(iii) ⇒ (i) are not valid if F is not a cusp form (cf. [Ma, p. 306]).

1.3. The second step of the Fourier expansion, the case of det(β)6= 0.

Next we consider the case where det(β) 6= 0. For each β ∈ Sym(2)Q withdet(β) 6= 0, the identity component (as an algebraic group) of the stabilizer of β

in M is given by Mβ = m(h, det(h)) | thβh = det(h)β. Let

kβ := Q[t]/(t2 + detβ) ∼=

Q(√−det β) −detβ 6∈ (Q×)2;

Q⊕Q −detβ ∈ (Q×)2,

be a quadratic separable algebra over Q. We set Akβ:= A ⊗Q kβ . Note that

there is an isomorphism Mβ∼= Reskβ/Q GL(1) or Mβ

∼= GL(1)×GL(1) accordingas −det β 6∈ (Q×)2 or −detβ ∈ (Q×)2. By Lemma 1.1, we have Fβ(γg) = Fβ(g)(γ ∈ Mβ,Q). In order to get an expansion of Fβ(g), we set

Ξω :=χ : Mβ,Q\Mβ,A → C(1) | character, χ(z) = ω(z) (∀z ∈ ZA)

,

Ξ0 :=χ : Mβ,Q\Mβ,A → C(1) | character, χ(z) = 1 (∀z ∈ ZA)

.

We define the subgroup Rβ of G by Rβ = Mβ nN. For each character χ ∈ Ξω,we define a character χ · ψβ of Rβ,A by

(χ · ψβ)(mn) = χ(m)ψβ(n), (m,n) ∈ Mβ,A ×NA.

We consider the following integral

Wχ·ψβ

F (g) :=∫

ZAMβ,Q\Mβ,A

Fβ(mg)χ(m)−1dm, g ∈ GA. (1.7)

1214 T. Moriyama

If the integral (1.7) converges absolutely, then we call Wχ·ψβ

F (g) the global gener-alized Whittaker function attached to F with respect to the character χ ·ψβ . Theglobal generalized Whittaker function W

χ·ψβ

F (g) belongs to the following space

C∞(Rβ,A\GA;χ · ψβ

)

:=

W : GAC∞→ C | W (rg) = (χ · ψβ)(r)W (g), ∀(r, g) ∈ Rβ,A ×GA

.

(1.8)

For a fixed Haar measure on ZAMβ,Q\Mβ,A, we denote by dχ the Haar measureon Ξ0 dual to it. Fix an arbitrary element χ1 of Ξω. Through the bijectionΞ0 3 χ 7→ χ · χ1 ∈ Ξω, we have a measure on Ξω, which is independent of thechoice of the base point χ1. The measure on Ξω obtained in this manner is alsodenoted by dχ. Then we have the following expansion of Fβ when det(β) 6= 0.

Proposition 1.3. Let F ∈ A (GQZA\GA;ω) be an automorphic form onGA with central character ω.

(i) Suppose that −det(β) 6∈ (Q×)2. Then ZAMβ,Q\Mβ,A is a compactabelian group and the integral (1.7) converges absolutely. Moreover if we nor-malize the Haar measure on ZAMβ,Q\Mβ,A so that the total volume is one, thenwe have the following inversion formula:

Fβ(g) =∫

Ξω

Wχ·ψβ

F (g)dχ =∑

χ∈Ξω

Wχ·ψβ

F (g). (1.9)

(ii) Suppose that −det(β) ∈ (Q×)2. If F is a cusp form, then the integral(1.7) converges absolutely. Moreover we have

Fβ(g) =∫

Ξω

Wχ·ψβ

F (g)dχ. (1.10)

Proof.

(i) Suppose that −det(β) 6∈ (Q×)2. Then the abelian group ZAMβ,Q\Mβ,A

is compact, for it is isomorphic to A×k×β \A×kβ

. Hence the measure dχ of Ξω is thepointing measure. This proves the inversion formula (1.9).

(ii) By Lemma 1.1, we may assume that β =(

0 11 0

). Then we have

Mβ = z diag(y, 1, 1, y) | y, z ∈ Gm.

We shall prove the convergence of the integral


∫

Q×\A×d×y

∫

NQ\NA

dn|F (n diag(y, 1, 1, y)g)| (1.11)

for each g ∈ GA. Since diag(y, 1, 1, y) commutes with n(0, 0, x2, 0), it suffices toshow that the integral

∫

Q×\A×d×y

∫

(Q\A)2dx1dx3

∣∣F (n(0, x1, 0, x3) diag(y, 1, 1, y)g)∣∣ (1.12)

converges absolutely for all g ∈ GA and defines a bounded function on eachcompact subset Ω of GA. Moreover, it is easy to see that

∫Q×\A×|y|≤1

d×y

∫

(Q\A)2dx1dx3

∣∣F (n(0, x1, 0, x3) diag(y, 1, 1, y)g)∣∣

=∫

Q×\A×|y|≥1

d×y

∫

(Q\A)2dx1dx3

∣∣F (n(0, x3, 0, x1) diag(y, 1, 1, y)w1g)∣∣.

Hence our assertion follows from Lemma 1.4 below. ¤

Lemma 1.4. Let F ∈ A cusp(GQZA\GA;ω) be a cusp form on GA withcentral character ω. Fix r ≥ 1 and a compact subset Ω of GA. Then there existsa constant C > 0 such that

|F (n(0, x1, 0, x3) diag(y, 1, 1, y)u)| ≤ C × |y|−rA ,

∀(x1, x3) ∈ A2, ∀y ∈ A× with |y|A ≥ 1, and ∀u ∈ Ω.

Proof (similar to [J-S, Lemma 3.4 (i)]). Let A0 := diag(a0a1, a0a2, a−11 ,

a−12 ) | ai ∈ Gm be the maximal Q-split torus in G. For ∞ ≥ t′ > t > 0, we set

A0,A(t′; t) :=

diag(a0a1, a0a2, a−11 , a−1

2 )

∈ A0,A

∣∣∣∣ t′ >

∣∣∣∣a1

a2

∣∣∣∣ > t and t′ > |a0a22| > t

.

By the compactness of KA · Ω, we have

KA · Ω ⊂ N0,A ·A0,A(t2; t1) ·KA

for some ∞ > t2 > t1 > 0. Since F is a cusp form with central character, it

1216 T. Moriyama

is rapidly decreasing on N0,AA0,A(∞; t)KA for every t > 0. That is, for each(r1, r2) ∈ R2 there exists a constant C > 0 such that

∣∣F (n diag(a0a1, a0a2, a

−11 , a−1

2 )k)∣∣ ≤ C

∣∣∣∣a1

a2

∣∣∣∣−r1∣∣a0a

22

∣∣−r2

∀n ∈ N0,A, ∀diag(a0a1, a0a2, a

−11 , a−1

2

) ∈ A0,A(∞; t), ∀k ∈ KA.

(1.13)

We shall rewrite the element n(0, x1, 0, x3) diag(y, 1, 1, y)u in order that we canapply the estimate (1.13) to it. By reduction theory for GL(2), there exists aconstant t0 > 0 independent of x3 and y such that

(1 x3y0 y

)= γ2

(1 x′30 1

)(b1 00 b2

)k2

for some γ2 ∈ GL(2)Q, b1, b2 ∈ A× with |b1/b2| ≥ t0, x′3 ∈ A, and k2 ∈ O(2) ·GL(2, Z). We also have

(y x1

0 1

)=

(det(γ2) 0

0 1

)(1 x′10 1

)(b1b2 00 1

)(det(k2) 0

0 1

)

with x′1 = det(γ2−1)x1. Hence we have

n(0, x1, 0, x3) diag(y, 1, 1, y) = γn(0, x′1, 0, x′3

)diag(b1b2, b1, 1, b2)k (1.14)

for some γ ∈ GQ and k ∈ KA.We first consider the case where b2 ∈ A× in (1.14) can be taken so that

|b2| ≥ 1. Then we have

n(0, x1, 0, x3) diag(y, 1, 1, y)u ∈ GQN0,AA0,A(∞; t3)KA

with t3 := t1 min1, t0. Hence for each (r1, r2) ∈ R2, there exists a constantC1 > 0 such that

F(n(0, x1, 0, x3) diag(y, 1, 1, y)u

) ≤ C1 × |b2|−r1

∣∣∣∣b1

b2

∣∣∣∣−r2

.

We set r1 = 2r and r2 = r. By noting |b1b2| = |y|, we have the estimate of thelemma in this case. Next we suppose that |b2| ≤ 1. In the expression


n(0, x1, 0, x3) diag(y, 1, 1, y) = γw1n(0, x′3, 0, x′1

)diag(b1, b1b2, b2, 1)w−1

1 k,

we note that γw1 ∈ GQ, diag(b1, b1b2, b2, 1) ∈ A0,A(∞; 1) and w−11 k ∈ KA. Hence

for each (r1, r2) ∈ R2 we can take a constant C2 > 0 such that

F(n(0, x1, 0, x3) diag(y, 1, 1, y)u

) ≤ C2 × |b2|r1 |b1b2|−r2 .

By putting r1 = 0 and r2 = r, we have the estimate of the lemma for |b2| ≤ 1,too. ¤

1.4. The case of cusp forms.Suppose that F ∈ A cusp(GQZA\GA;ω) is a cusp form on GSp(2)A. Then,

combining the results of the previous two subsections, we have the following Fourierexpansion of F :

F (g) =∑

γ∈B′Q\MQ

Fβ(1)(γg) +12

∑

d∈(Q×)2\Q×

∑

γ∈Mβ(2)d

,Q\MQ

Fβ

(2)d

(γg)

=∑

γ∈B′Q\MQ

∑

α∈Q×WF (diag(α, 1, α−1, 1)γg)

+12

∑

d∈(Q×)2\Q×

∑

γ∈Mβ(2)d

,Q\MQ

∫

Ξω

Wχ·ψ

β(2)d

F (γg)dχ.

Here B′ :=m

(( a11 a120 a22

), a2

22

) ∈ M

is the stabilizer of β(1) in M. Note that thefactor 1/2 comes from the fact that Mβ,Q is of index two in the stabilizer of β inMQ.

Remark. The above expansion plus Proposition 1.2 implies that for a non-zero cusp form F we can find β and χ such that the associated global generalizedWhittaker function W

χ·ψβ

F (γg) does not vanish. This is quite satisfactory, becausewe can study the spinor L-function by the method of Andrianov [An], [PS] withoutassuming any global conditions on F .

1.5. Local generalized Whittaker functions.We introduce the local generalized Whittaker functions, which are local coun-

terpart of the global generalized Whittaker functions Wχ·ψβ

F (g). For a place v of Q,we denote by ψv the restriction of the character ψ to Qv. For each β ∈ Sym(2)Qv

,we define a character ψv,β of NQv

by

1218 T. Moriyama

ψv,β

((I2 x

02 I2

))= ψv(tr(βx)).

As before, we define Mβ to be the identity component of the stabilizer of β in Mand set Rβ := Mβ nN, which are algebraic subgroups of G defined over Qv. Foreach quasi-character χ : Mβ,Qv

→ C×, we set

C∞(Rβ,Qv

\GQv;χ · ψv,β

)

:=

W : GQv

C∞−→ C | W (rg) = (χ · ψv,β)(r)W (g), ∀(r, g) ∈ Rβ,Qv×GQv

,

on which GQvacts by right translation.

We now suppose that v = ∞ is the real place and set Rβ := Rβ,R. Recall thata C∞-function f : G → C is said to be of moderate growth if there exist constantsC > 0 and M > 0 such that |f(g)| < C‖g‖M

∞ holds for all g ∈ G. If f and its allderivatives f(g;X) (X ∈ U(g)) are of moderate growth with an exponent M > 0common to all X ∈ U(g), then the function f is said to be of uniformly moderategrowth. We denote by C∞mg(Rβ\G;χ · ψβ) (resp. C∞umg(Rβ\G;χ · ψβ)) the totalityof functions in C∞(Rβ\G;χ·ψβ) of moderate growth (resp. of uniformly moderategrowth). Note that an automorphic form F ∈ A (ZAGQ\GA;ω) is of moderategrowth in the sense that there exist C > 0 and M > 0 such that

|F (g)| ≤ C × ‖g‖M , ∀g ∈ GA.

Hence, for each fixed gf ∈ GAf , the function Wχ·ψβ

F (gfg∞) in g∞ ∈ GR belongsto the space C∞mg(Rβ\G;χ · ψβ). Hence we are led to the following definition.

Definition 1.5. Let (π, Hπ) be a quasi-simple (g,K)-module.

( i ) By a generalized Whittaker functional belonging to π, we understand anelement of the intertwining space

GWG(π, χ · ψβ) := Homg,K

(Hπ, C∞(Rβ\G;χ · ψβ)

).

( ii ) If Φ ∈ GWG(π, χ · ψβ) belongs to the subspace

GWmgG (π, χ · ψβ) := Homg,K


),

then we say that it has the moderate growth property.

It is preferable that the multiplicity free property dimC GWmgG (π, χ ·ψβ) ≤ 1


holds for an arbitrary irreducible (g,K)-module (π, Hπ) and an arbitrary quasi-character χ · ψβ . One of the main purposes of this paper is to prove that thisexpectation is true when det(β) < 0 and (π, Hπ) is equivalent to one of two kindsof irreducible (g,K)-modules introduced in the next section.

Before closing this section, we mention some results on the generalized Whit-taker models in the non-archimedean case. In the first place, Bump, Friedberg,and Furusawa [Bu-Fr-Fu] obtained an explicit formula of the unramified gener-alized Whittaker function by the method of [C-S]. This is the ultimate resultextending an earlier result of Andrianov [An, Theorem 1] (cf. [Su, Proposition2-5]). Some results related to the multiplicity freeness are proved in [No], [No-PS],where the authors consider the stabilizer group of ψβ instead of its connectedcomponent Mβ . Moreover F. Rodier [Ro] proved the following multiplicity freetheorem for the cuspidal representations of GQp :

Proposition 1.6 ([Ro, Theorem, p. 126]). Suppose that v = p < ∞ isa finite place of Q. Let (π, Hπ) be a cuspidal irreducible admissible representa-tion of GQp with trivial central character. Then the intertwining space HomGQp

(π, C∞(Rβ,Qp\GQp ;χ · ψβ)) is at most one dimensional.

2. The group Sp(2,R) and its representations.

In this section, we fix some notation concerning the Lie group Sp(2,R) andintroduce two kinds of standard representations of it.

2.1. Root systems and the irreducible K0-modules.The maximal compact subgroup K0 of G0 = Sp(2,R) is isomorphic to the

unitary group U(2) := g ∈ GL(2,C) | tgg = I2 of degree two. Fix an isomor-phism κ : U(2) ∼= K0 by

κ : U(2) 3 A +√−1B 7→ kA,B :=

(A B−B A

)∈ K0, (A,B ∈ Mat(2)R).

The differential κ∗ of κ defines an isomorphism of Lie algebras: κ∗ : gl(2,C) ∼= kC .A compact Cartan subalgebra of g0 is given by h := RT1 ⊕RT2, where

T1 := κ∗

((√−1 00 0

)), T2 := κ∗

((0 00√−1

)).

Define a C-basis e1, e2 of h∗C by ei(Tj) =√−1δij (1 ≤ i, j ≤ 2). Then the

root system ∆ = ∆(g0,C , hC) for the pair (g0,C , hC) is given by ∆(g0,C , hC) =±2e1,±2e2,±(e1 ± e2). We denote by ∆c (resp. ∆nc) the set of compact roots

1220 T. Moriyama

(resp. the set of non-compact roots) in ∆: ∆c = ±(e1−e2) (resp. ∆nc = ∆\∆c).We take a positive system ∆+ of ∆ as ∆+ := 2e1, e1 + e2, 2e2, e1− e2. Then thesets of compact and non-compact positive roots in ∆ are given by ∆+

c = ∆c ∩∆+

and ∆+nc = ∆nc ∩∆+, respectively. For each symmetric matrix A ∈ Sym(2)C , we

define elements p±(A) of g0,C by

p±(A) :=(

A ±√−1A

±√−1A −A

)∈ g0,C .

Then we can take the root vectors X(α1,α2) ∈ g0,C corresponding to the non-compact roots α1e1 + α2e2 = (α1, α2) ∈ ∆nc as follows:

X±(2,0) := p±

((1 00 0

)), X±(1,1) := p±

((0 11 0

)), X±(0,2) := p±

((0 00 1

)).

We set p± :=⊕

α∈∆+nc

CX±α.The set of all the irreducible finite-dimensional representations of K0 is pa-

rameterized by their highest weights relative to ∆+c . For each dominant integral

weight q = (q1, q2) = q1e1 + q2e2 ∈ h∗C (qi ∈ Z, q1 ≥ q2), we denote the corre-sponding irreducible finite-dimensional representation by (τ(q1,q2), V(q1,q2)). Thedimension of the representation space V(q1,q2) is given by d + 1, where we setd = dq = q1 − q2. There is a basis vk | 0 ≤ k ≤ d of (τ(q1,q2), V(q1,q2)) satisfying

τq

(κ∗

((1 00 0

)))vk = (k + q2)vk, τq

(κ∗

((0 00 1

)))vk = (−k + q1)vk,

τq

(κ∗

((0 10 0

)))vk = (k + 1)vk+1, τq

(κ∗

((0 01 0

)))vk = (d− k + 1)vk−1,

which we call the standard basis of (τ(q1,q2), V(q1,q2)). Here we understand thatv−1 = vd+1 = 0. Note that (Ad, p+) and (Ad, p−) are equivalent to τ(2,0) andτ(0,−2), respectively. The correspondence of the bases are given by

(X(2,0), X(1,1), X(0,2)) 7→ (v2, v1, v0), (X(0,−2), X(−1,−1), X(−2,0)) 7→ (v2,−v1, v0).

The simple Lie algebra g0 has a R-split Cartan subalgebra a := RH1⊕RH2,where we set

H1 := diag(1, 0,−1, 0), H2 := diag(0, 1, 0,−1).


We denote the basis of a∗C dual to H1,H2 by e′1, e′2. Then the root systemΣ = Σ(g0, a) for (g0, a) is given by ±2e′1,±2e′2,±(e′1±e′2). As a positive system ofΣ, we take Σ+ := 2e′1, 2e′2, e

′1+e′2, e

′1−e′2. For each α = (α1, α2) = α1e

′1+α2e

′2 ∈

Σ+, we fix a root vector Eα as

E(2,0) := n∗

((1 00 0

)), E(1,1) := n∗

((0 11 0

)),

E(0,2) := n∗

((0 00 1

)), E(1,−1) :=

0 1 0 00 0 0 00 0 0 00 0 −1 0

.

Here we set n∗(A) :=( 02 A

02 02

) ∈ g0 for A ∈ Sym(2)R. For a negative root −α

(α ∈ Σ+), we fix a root vector E−α as E−α := tEα.

2.2. Standard representations of Sp(2,R).In this subsection, we introduce two kinds of quasi-simple admissible repre-

sentations of Sp(2,R).

2.2.1. P1-principal series representations.We define the non-Siegel maximal parabolic subgroup P1 of Sp(2,R) to be the

stabilizer of the line R·t(1, 0, 0, 0) in Sp(2,R). We fix the Langlands decompositionP1 = M1A1N1 of P1 as follows:

M1 :=

ε1a b

cε1

d

∣∣∣∣∣∣∣∣ε1 = ±1,

(a bc d

)∈ SL(2,R)

,

A1 :=

diag(a1, 1, a−11 , 1) | a1 > 0

, N1 := n(x0, x1, x2, 0) | xi ∈ R.

Let D±n (n ≥ 1) be the (limit of) discrete series representation of SL(2,R) with

Blattner parameter±n. Then we have an irreducible unitary representation (σ, Vσ)of M1 characterized by σ(diag(−1, 1,−1, 1)) = ε (ε = ±1) and σ|SL(2,R)

∼= D±n ,

which we denote by σ = ε⊗D±n . For ν1 ∈ C, we define a quasi-character exp(ν1) :

A1 → C× by

exp(ν1)(diag(a1, 1, a−1

1 , 1))

= aν11 .

Then the P1-principal series representation I(P1;σ, ν1) of G0 is realized on the

1222 T. Moriyama

space of all C∞-functions f : G0 −→ Vσ satisfying

f(mang) = σ(m) exp(ν1 + 2)(a)f(g),

∀(m,a, n, g) ∈ M1 ×A1 ×N1 ×G0,

on which G0 acts by right translation. The infinitesimal character of π is givenby ν1e

′1 + (n − 1)e′2 ∈ a∗C/ ∼W , where W = W (g0, a) is the Weyl group for the

pair (g0, a). For an admissible representation (π, Hπ) of G0 and an irreduciblefinite-dimensional representation τ of K0, we set [π; τ ] := dimC HomK0(τ, π), themultiplicity of τ in π. From [Mi-1, Proposition 6.3], we quote the following:

Proposition 2.1. Let π = I(P1;σ, ν1) be a P1-principal series representa-tion of G0. Then we have the following assertions:

( i ) if σ = (−1)n ⊗D+n , then we have [π; τ(n,n)] = 1 and [π; τ(n−2,n−2)] = 0;

( ii ) if σ = (−1)n+1⊗D+n , then we have [π; τ(n,n−1)] = 1 and [π; τ(n−1,n−2)] = 0;

(iii) if σ = (−1)n ⊗D−n , then we have [π; τ(−n,−n)] = 1 and [π; τ(2−n,2−n)] = 0;

(iv) if σ = (−1)n+1⊗D−n , then we have [π; τ(1−n,−n)] = 1 and [π; τ(2−n,1−n)] = 0.

We say that a P1-principal series representation π = I(P1;σ, ν1) of G0 is even(resp. odd) if σ is of the form σ = (−1)n ⊗D±

n (resp. σ = (−1)n+1 ⊗D±n ). We

denote by I(P1;σ, ν1)[c] (c ∈ C) the representation π of GSp(2,R) characterizedby

π|G0∼= I(P1;σ, ν1)⊕ I(P1;σ∨,−ν1) and π(zI4) = zc (∀z > 0).

Here σ∨ stands for the contragradient representation of σ.

2.2.2. (Limits of) large discrete series representations.Let D(λ1,λ2) be the (limit of) large discrete series representation of Sp(2,R)

with minimal K0-type τ(λ1,λ2), where (λ1, λ2) ∈ Z⊕2 satisfies 1 − λ1 ≤ λ2 ≤ 0 or1 + λ2 ≤ −λ1 ≤ 0 (cf. [Kn, Theorem 9.20, Theorem 12.26, and p. 626–627], [Mo,Subsection (1.2)]). The infinitesimal character of π is given by (λ1− 1)e1 +λ2e2 ∈a∗C/ ∼W . We also note the following:

Proposition 2.2. Let π = D(λ1,λ2) be a (limit of ) large discrete seriesrepresentation of G0 with 1− λ1 ≤ λ2 ≤ 0. Then the minimal K0-type τ(λ1,λ2) ofπ occurs in π with multiplicity one. Moreover, if an irreducible finite-dimensionalrepresentation τ of K0 occurs in π, then τ is equivalent to τ(q1,q2) with

(q1, q2) = (λ1, λ2) + k(1, 1) + l(0,−2) for some k, l ∈ Z≥0.


For each (λ1, λ2) ∈ Z⊕2 satisfying 1 − λ1 ≤ λ2 ≤ 0 and c ∈ C, there existsan irreducible admissible representation π of GSp(2,R) characterized by

π |Sp(2,R)= D(λ1,λ2) ⊕D(−λ2,−λ1) and π(zI4) = zc, (∀z > 0),

which we denote by π = D(λ1,λ2)[c].

3. Generalized Whittaker functions on Sp(2,R).

We introduce the generalized Whittaker functions on G0 = Sp(2,R) andprove their elementary properties. The relation with the generalized Whittakerfunctions on G = GSp(2,R) will be discussed in Subsection 8.1.

3.1. Definition of local generalized Whittaker functions onSp(2,R).

We fix a symmetric matrix β ∈ Sym(2)R satisfying det(β) 6= 0. Let

Tβ := t = m(h, 1) ∈ MR | thβh = β

be the stabilizer of β in MR ∩ G0. Then the identity component (with respectto the Euclidean topology) T β of Tβ is isomorphic to SO(2) or R×

>0 according asdet(β) > 0 or det(β) < 0. We set R1

β := T β nNR. For a quasi-character χ : T β →C×, we define a quasi-character χ·ψβ : R1

β → C× of R1β by (χ·ψβ)(tn) = χ(t)ψβ(n)

(t ∈ T β , n ∈ NR). Then we have a smooth representation induced from χ · ψβ :

C∞(R1

β\G0;χ · ψβ

)

:=W : G0

C∞−→ C | W (rg) = (χ · ψβ)(r)W (g), ∀(r, g) ∈ R1β ×G0

,

on which G0 acts by right translation. As in Subsection 1.5, a function f : G0 → C

is said to be of moderate growth if there exists a constant C > 0 and M > 0such that |f(g)| ≤ C‖g‖M for all g ∈ G0. If f and its all derivatives f(g;X)(X ∈ U(g0)) are of moderate growth with exponent M > 0 common to all X ∈U(g0), then we say that the function f is of uniformly moderate growth. Wedenote by C∞mg(R

1β\G0;χ ·ψβ) (resp. C∞umg(R

1β\G0;χ ·ψβ)) the space of functions

f ∈ C∞(R1β\G0;χ ·ψβ) of moderate growth (resp. of uniformly moderate growth).

Definition 3.1. Let (π, Hπ) be a quasi-simple (g0,K0)-module.

( i ) By a generalized Whittaker functional, we understand an element of theintertwining space

1224 T. Moriyama

GWG0(π, χ · ψβ) := Homg0,K0

(Hπ, C∞(R1

β\G0;χ · ψβ)).

For a vector v ∈ Hπ, we call its image Φ(v) under some Φ ∈ GWG0(π, χ·ψβ)a generalized Whittaker function on G0 belonging to π.

( ii ) If Φ ∈ GWG0(π, χ · ψβ) belongs to the subspace

GWmgG0

(π, χ · ψβ) := Homg0,K0

(Hπ, C∞mg(R

1β\G0;χ · ψβ)

),

then we say that Φ and Φ(v) have the moderate growth property.(iii) For a non-zero element Φ ∈ GWG0(π, χ · ψβ), we call the whole image

Φ(Hπ) a generalized Whittaker model of π. If Φ ∈ GWmgG0

(π, χ · ψβ), thenwe say Φ(Hπ) has the moderate growth property.

Remark.

( i ) In the literature, generalized Whittaker functions are sometimes calledBessel functions (e.g. [F], [TB], [Pr-TB], [Pi-Sch]), Siegel-Whittaker func-tions (e.g. [Is]), or generalized Bessel functions (e.g. [No], [No-PS]).

( ii ) Since the generalized Whittaker function Φ(v) is right K0-finite and Z(g0)-finite, it is a real analytic function on G0 by the elliptic regularity theorem.

Let β, β′ ∈ Sym(2)R be two symmetric matrices with det(β) 6= 0 anddet(β′) 6= 0. Suppose that there exists an element m0 = m(h, 1) ∈ MR ∩ G0

such that β · m0 = thβh = β′. Define a quasi-character χ′ of T β′ = m−10 T β m0

by χ′(t′) := χ(m0t′m−1

0 ) (t′ ∈ T β′). Then we have the following isomorphism ofG0-modules

C∞(R1

β\G0;χ · ψβ

) ∼= C∞(R1

β′\G0;χ′ · ψβ′), (3.1)

which assigns W (g) to W (m0g). For a finite-dimensional K0-module (τ, Vτ ), weset

C∞(R1

β\G0/K0;χ · ψβ ; τ)

:=W : G0

C∞−→ V ∨τ | W (rgk) = (χ · ψβ)(r)τ∨(k)−1f(g),

∀(r, g, k) ∈ R1β ×G0 ×K0

,

where (τ∨, V ∨τ ) is the contragradient representation of (τ, Vτ ). Fix a K0-

equivariant map ιτ : Vτ → Hπ. For a generalized Whittaker functional Φ ∈GWG0(π, χ·ψβ), we have a V ∨

τ -valued C∞-function W ∈ C∞(R1β\G0/K0;χ·ψβ ; τ)

characterized by


Φ(ιτ (v))(g) = 〈W (g), v〉, ∀g ∈ G0, ∀v ∈ Vτ .

Here 〈· , ·〉 stands for the canonical pairing between V ∨τ and Vτ . We call the

function W ∈ C∞(R1β\G0/K0;χ ·ψβ ; τ) defined from Φ in this way the generalized

Whittaker function on G0 of type (π, χ · ψβ , ιτ ). If τ is irreducible and occursin π with multiplicity one, then we simply say that f is of type (π, χ · ψβ , τ). IfΦ ∈ GWmg

G0(π, χ·ψβ), then we say that the corresponding W ∈ C∞(R1

β\G0/K0;χ·ψβ ; τ) has the moderate growth property.

3.2. Radial parts.From now on, we shall assume that det(β) < 0. In view of (3.1), we may

suppose that β =( 0 c/2

c/2 0

)(c ∈ R×). We may further assume that c = 1, but we

do not assume it. It is easy to check that

T β =

diag(√

y1,1√y1

,1√y1

,√

y1

) ∣∣∣∣ y1 > 0

.

For (t, y) ∈ R×R>0, we set

a(t, y) := exp

t · H0

2+ log(y) · (H1 + H2)

2

with H0 := E(1,−1) + E(−1,1) ∈ g0.

Then we have

a(t, y) =

y1/2 ch(t/2) y1/2 sh(t/2)y1/2 sh(t/2) y1/2 ch(t/2)

y−1/2 ch(t/2) y−1/2 sh(−t/2)y−1/2 sh(−t/2) y−1/2 ch(t/2)

.

Here we use the abbreviated expressions

ch(t) := cosh(t), sh(t) := sinh(t), and th(t) := tanh(t).

We define a closed abelian subgroup S of G0 by

S := a(t, y) | t ∈ R, y > 0.

Then we have the following:

Lemma 3.2. The multiplication map R1β × S ×K0 3 (r, a, k) 7→ rak ∈ G0

gives a diffeomorphism R1β × S ×K0

∼= G0.

1226 T. Moriyama

Proof. First we prove the surjectivity of the multiplication map. Considerthe usual action of SL(2,R) on the upper half plane H1 := z ∈ C | Im(z) > 0:g · 〈z〉 := (az + b)/(cz + d)

(g =

(a bc d

), z ∈ H1

). From the equality

√

y1 0

01√y1

(ch(t/2) sh(t/2)sh(t/2) ch(t/2)

)· 〈√−1〉 = y1

th(t)+

√−1ch(t)

, (y1 > 0, t ∈ R),

we know that MR ∩ G0 = T β S(MR ∩ K0). Since G0 = NR(MR ∩ G0)K0, thisproves R1

βSK0 = G0. To prove the injectivity, it is enough to show that ra1 = a2k

for some r ∈ R1β , a1, a2 ∈ S, and k ∈ K0 implies r = k = I4 and a1 = a2, the proof

of which is an easy computation. Finally, Lemma 4.2 in the next section tells usthat

g0 = Ad(a−1) Lie(R1

β

)⊕ Lie(S)⊕ k, ∀a ∈ S. (3.2)

This decomposition implies that the multiplication map is a diffeomorphism. ¤

Let C∞(S; τ) be the space of V ∨τ -valued C∞-functions on S. It follows from

Lemma 3.2 that the restriction map

C∞(R1

β\G0/K0;χ · ψβ ; τ) → C∞(S; τ)

is an isomorphism. We call the restriction W|S(a(t, y)) of W (g) ∈C∞(R1

β\G0/K0;χ · ψβ ; τ) to S the S-radial part of W (g). A well-known theo-rem of Harish-Chandra [HC, Theorem 1] (see also [Bo, 5.6], [Bu, Section 2.10])tells us that a C∞-function f : G0 → C of moderate growth is of uniformly mod-erate growth if it is right K0-finite and Z(g0)-finite. Hence we have the followingisomorphism

GWmgG0

(π, χ · ψβ) ∼= Homg0,K0

(Hπ, C∞umg

(R1

β\G0;χ · ψβ

))(3.3)

for a quasi-simple (g0,K0)-module (π, Hπ). The following lemma will play a cru-cial role in the proof of our main results.

Lemma 3.3. Suppose that W (g) ∈ C∞umg(R1β\G0;χ · ψβ). For each N ≥ 0,

there exists a constant C > 0 such that |W (a(0, y))| ≤ Cy−N for all y > 0.

Proof. Since W (g) is assumed to be of uniformly moderate growth, thereexists a constant N > 0 such that for each l ≥ 0 we can find Cl > 0 satisfying


∣∣W (a(0, y);El(1,1))

∣∣ ≤ Cl × (maxy, y−1)N , ∀l ≥ 0, ∀y > 0.

On the other hand, we have

W(a(0, y);E(1,1)

)

=d

ds

∣∣∣∣s=0

W(a(0, y) exp(sE(1,1))

)

=d

ds

∣∣∣∣s=0

W(exp(syE(1,1))a(0, y)

)= 2π

√−1cyW (a(0, y)). (3.4)

Hence we have the assertion of the lemma. ¤

4. Differential operators.

In this section we introduce two kinds of differential operators, that is, the shiftoperators and the Casimir operator, which will be used to construct differentialequations satisfied by the generalized Whittaker functions in Section 5–Section 7.

4.1. Schmid operators.In this subsection, we introduce the Schmid operators, which are used to

define shift operators in the next subsection. Let (τ, Vτ ) be a finite-dimensionalrepresentation of K0. Recall that β =

( 0 c/2c/2 0

). For µ ∈ C, we define a quasi-

character χµ of T β by

χµ

(diag

(√y1,

1√y1

,1√y1

,√

y1

))= yµ

1 , y1 > 0. (4.1)

For a K0-equivariant map

φτ ∈ HomK0

(τ, C∞

(R1

β\G0;χµ · ψβ

)) ∼= C∞(R1

β\G0/K0;χµ · ψβ ; τ),

we define K0-equivariant maps φp±⊗τ : p± ⊗ Vτ → C∞(R1β\G0;χ · ψβ) by

φp±⊗τ (X ⊗ v)(g) := φτ (v)(g;X), (X ∈ p±, v ∈ Vτ , g ∈ G0). (4.2)

The assignment of φτ to φp±⊗τ defines the gradient type differential operators

∇± : C∞(R1

β\G0/K0;χµ · ψβ ; τ) → C∞

(R1

β\G0/K0;χµ · ψβ ; τ ⊗Adp±).

1228 T. Moriyama

These operators ∇± are called the Schmid operators. By the identification (τ ⊗Adp±)∨ ∼= τ∨⊗Adp∓ , we regard the image [∇±W ](g) of W ∈ C∞(R1

β\G0/K0;χµ ·ψβ ; τ) as a (V ∨

τ ⊗ p∓)-valued C∞-function on G0. By using the root vectors Xα

(α ∈ ∆nc) introduced in Section 2, we can express [∇±W ](g) as follows:

[∇±W ](g) = W(g;X±(2,0)

)⊗X∓(2,0) +12W

(g;X±(1,1)

)⊗X∓(1,1)

+ W(g;X±(0,2)

)⊗X∓(0,2).

The S-radial parts of the actions of the Schmid operators ∇± can be described asfollows:

Proposition 4.1. Suppose that W ∈ C∞(R1β\G0/K0;χµ · ψβ ; τ). Then we

have

[∇+W ](a(t, y))

=

δy − sh(t) · (2πcy) +µ

ch(t)+ (τ∨ ⊗Ad)

κ∗

−1th(t)

2− th(t)

20

− 3

·W (a(t, y))⊗X(−2,0)

+

∂t − ch(t) · (2πcy) + th(t)− 12(τ∨ ⊗Ad)

(κ∗

((0 11 0

)))

·W (a(t, y))⊗X(−1,−1)

+

δy − sh(t) · (2πcy)− µ


κ∗

0 − th(t)2

th(t)2

−1

− 3

·W (a(t, y))⊗X(0,−2)

and

[∇−W ](a(t, y))

=

δy + sh(t) · (2πcy) +µ


κ∗

1th(t)

2− th(t)

20

− 3

·W (a(t, y))⊗X(2,0)


+

∂t + ch(t) · (2πcy) + th(t) +12(τ∨ ⊗Ad)

(κ∗

((0 11 0

)))

·W (a(t, y))⊗X(1,1)

+

δy + sh(t) · (2πcy)− µ


κ∗

0 − th(t)2

th(t)2

1

− 3

·W (a(t, y))⊗X(0,2).

Here we use the abbreviated expressions:

∂t :=∂

∂t, δy := y

∂

∂y. (4.3)

Proof. For any function W (g) ∈ C∞(R1β\G0/K0;χ · ψβ ; τ), we have the

following equality

W

(a(t, y); Ad(a(t, y)−1)(ξ) ·

(H0

2

)l

·(

H1 + H2

2

)m

· η)

= (χ · ψβ)(ξ)τ∨(η)∂ltδ

my W (a(t, y)), (4.4)

∀ξ ∈ U(Lie(R1β)), ∀l, ∀m,≥ 0, ∀η ∈ U(k).

Here U(k) 3 η 7→ η ∈ U(k) is the anti-automorphism of U(k) characterized byX = −X for X ∈ kC . By using (4.4) and Lemma 4.2 below, we can compute∇±W (a(t, y)). ¤

Lemma 4.2. For each a(t, y) ∈ S, the root vectors Xα (α ∈ ∆nc) are de-composed as follows:

X±(2,0)

= Ad(a(t, y)−1)± 2

√−1y · n∗((

ch2(t/2) sh(t)/2sh(t)/2 sh2(t/2)

))+

12 ch(t)

(H1 −H2)

+12(H1 + H2) + κ∗

(( ±1 − th(t)/2th(t)/2 0

)),

1230 T. Moriyama

X±(1,1)

= Ad(a(t, y)−1)± 2

√−1y · n∗((

sh(t) ch(t)ch(t) sh(t)

))+ H0 ± κ∗

((0 11 0

)),

X±(0,2)

= Ad(a(t, y)−1)± 2

√−1y · n∗((

sh2(t/2) sh(t)/2sh(t)/2 ch2(t/2)

))− 1

2 ch(t)(H1 −H2)

+12(H1 + H2) + κ∗

((0 th(t)/2

− th(t)/2 ±1

)).

Remark. In [Mi-1], the corresponding computation is done for β =(h1 00 h2

), h1h2 6= 0. However, our anti-diagonal choice of β makes our final for-

mulae in Theorem 7.1 simple as long as we use the standard basis introduced inSubsection 2.2. This can be foreseen from the computation of local Novodvorskyintegrals in [Mo, Proposition 8] (see Section 9).

4.2. Shift operators.Suppose that (τ, Vτ ) is equivalent to an irreducible finite-dimensional repre-

sentation (τ(λ1,λ2), V(λ1,λ2)) of K0. The tensor product representation V ∨τ ⊗p± has

the decomposition into irreducible factors:

V ∨τ ⊗ p+

∼=

V(−λ2+2,−λ1) ⊕ V(−λ2+1,−λ1+1) ⊕ V(−λ2,−λ1+2) if λ1 > λ2;

V(−λ2+2,−λ1) if λ1 = λ2;

V ∨τ ⊗ p− ∼=

V(−λ2,−λ1−2) ⊕ V(−λ2−1,−λ1−1) ⊕ V(−λ2−2,−λ1) if λ1 > λ2;

V(−λ2,−λ1−2) if λ1 = λ2.

Here we understand that V(q1,q2) = 0 if q1 < q2. Let Pup, P ev, and P dn be theprojectors from V ∨

τ ⊗ p+ (resp. V ∨τ ⊗ p−) to V(−λ2+2,−λ1) (resp. V(−λ2,−λ1−2)),

V(−λ2+1,−λ1+1) (resp. V(−λ2−1,−λ1−1)), and V(−λ2,−λ1+2) (resp. V(−λ2−2,−λ1)),which are determined up to constant multiples. Then we have the linear maps

P • · ∇+ : C∞(R1

β\G0/K0;χ · ψβ ; τ) → C∞

(R1

β\G0/K0;χ · ψβ ; τ ′),

where τ ′ is τ(λ1+2,λ2), τ(λ1+1,λ2+1), or τ(λ1,λ2+2) according as • = up, ev, or dn.Similarly we have P • · ∇− (• = up, ev, or dn). We call these six differentialoperators P • ·∇± the shift operators. We write W (g) ∈ C∞(R1

β\G0;χ·ψβ ; τ(λ1,λ2))in the form


W (g) =d∑

k=0

φk(g)vk

with the standard basis vk | 0 ≤ k ≤ d = λ1 − λ2 of τ(−λ2,−λ1). We frequentlywrite φk(t, y) in place of φk(a(t, y)). In terms of the coefficient functions φk(t, y),the S-radial parts of the shift operators are described as follows:

Proposition 4.3.

( i ) We define C∞-functions φ(2,0)k (g) (0 ≤ k ≤ d+2) on G0 by [Pup·∇+W ](g) =∑d+2

k=0 φ(2,0)k (g)vk. Then we have

φ(2,0)k (t, y) = −1

4(d + 2− k)(d + 1− k)(d− k) th(t)φk+1(t, y)

+(

d + 2− k

2

)(δy − 2πcy · sh(t) +

µ

ch(t)+ λ1

)φk(t, y)

+ k(d + 2− k)(− ∂t + 2πcy · ch(t) +

d

4· th(t)

)φk−1(t, y)

+(

k

2

)(δy − 2πcy · sh(t)− µ

ch(t)+ λ1

)φk−2(t, y)

− 14k(k − 1)(k − 2) th(t)φk−3(t, y), (0 ≤ k ≤ d + 2).

( ii ) We define C∞-functions φ(1,1)k (g) (0 ≤ k ≤ d) on G0 by [P ev · ∇+W ](g) =∑d


φ(1,1)k (t, y) =

12(d− k)(d− k − 1) th(t)φk+2(t, y)

+ (k − d)(

δy − 2πcy · sh(t) +µ

ch(t)+

λ1 + λ2

2− 1

)φk+1(t, y)

+ (2k − d)(

∂t − 2πcy · ch(t) +12· th(t)

)φk(t, y)

+ k

(δy − 2πcy · sh(t)− µ

ch(t)+

λ1 + λ2

2− 1

)φk−1(t, y)

− 12k(k − 1) th(t)φk−2(t, y), (0 ≤ k ≤ d).

1232 T. Moriyama

(iii) We define C∞-functions φ(0,2)k (g) (0 ≤ k ≤ d−2) on G0 by [P dn·∇+W ](g) =∑d−2


φ(0,2)k (t, y) = −1

2(d− k − 2) th(t)φk+3(t, y)

+(

δy − 2πcy · sh(t) +µ

ch(t)+ λ2 − 1

)φk+2(t, y)

+ 2(

∂t − 2πcy · ch(t) +d + 2

4· th(t)

)φk+1(t, y)

+(

δy − 2πcy · sh(t)− µ

ch(t)+ λ2 − 1

)φk(t, y)

− k

2th(t)φk−1(t, y), (0 ≤ k ≤ d− 2).

(iv) We define C∞-functions φ(0,−2)k (g) (0 ≤ k ≤ d + 2) on G0 by [Pup ·

∇−W ](g) =∑d+2

k=0 φ(0,−2)k (g)vk. Then we have

φ(0,−2)k (t, y) =

14(d + 2− k)(d + 1− k)(d− k) th(t)φk+1(t, y)

+(

d + 2− k

2

)(δy + 2πcy · sh(t)− µ

ch(t)− λ2

)φk(t, y)

+ k(d + 2− k)(

∂t + 2πcy · ch(t)− d

4· th(t)

)φk−1(t, y)

+(

k

2

)(δy + 2πcy · sh(t) +

µ

ch(t)− λ2

)φk−2(t, y)

+14k(k − 1)(k − 2) th(t)φk−3(t, y), (0 ≤ k ≤ d + 2).

( v ) We define C∞-functions φ(−1,−1)k (g) (0 ≤ k ≤ d) on G0 by [P ev ·∇−W ](g) =∑d

k=0 φ(−1,−1)k (g)vk. Then we have

φ(−1,−1)k (t, y) = −1

2(d− k)(d− k − 1) th(t)φk+2(t, y)

+ (k − d)(

δy + 2πcy · sh(t)− µ

ch(t)− λ1 + λ2

2− 1

)φk+1(t, y)


+ (d− 2k)(

∂t + 2πcy · ch(t) +12· th(t)

)φk(t, y)

+ k

(δy + 2πcy · sh(t) +

µ

ch(t)− λ1 + λ2

2− 1

)φk−1(t, y)

+12k(k − 1) th(t)φk−2(t, y), (0 ≤ k ≤ d).

(vi) We define C∞-functions φ(−2,0)k (g) (0 ≤ k ≤ d − 2) on G0 by [P dn ·

∇−W ](g) =∑d−2

k=0 φ(−2,0)k (g)vk. Then we have

φ(−2,0)k (t, y) =

12(d− k − 2) th(t)φk+3(t, y)

+(

δy + 2πcy · sh(t)− µ

ch(t)− λ1 − 1

)φk+2(t, y)

+ (−2)(

∂t + 2πcy · ch(t) +d + 2

4· th(t)

)φk+1(t, y)

+(

δy + 2πcy · sh(t) +µ

ch(t)− λ1 − 1

)φk(t, y)

+k

2th(t)φk−1(t, y), (0 ≤ k ≤ d− 2).

Proof. We can prove these formulae in the same manner as [Mi-1, Propo-sition 10.2]. That is, we combine Proposition 4.2 with the formulae of projectorsP • given in [Mi-1, Lemmas 3.3, 3.4, and 3.5]. Details are left to the reader. ¤

4.3. The Casimir operator.Up to a constant multiple, the Casimir element Ω ∈ U(g0) of g0 is given by

Ω = H21 + H2

2 − 4H1 − 2H2 + 4E(2,0) · E(−2,0)

+ 2E(1,1) · E(−1,−1) + 4E(0,2) · E(0,−2) + 2E(1,−1) · E(−1,1).

For each finite-dimensional representation (τ, Vτ ) of K0, we call the differentialoperator

RΩ : C∞(R1

β\G0/K0;χ · ψβ ; τ) → C∞

(R1

β\G0/K0;χ · ψβ ; τ)

the Casimir operator. Then the S-radial part of the Casimir operator RΩ is givenby the following proposition:

1234 T. Moriyama

Proposition 4.4. For each W ∈ C∞(R1β\G0/K0;χ · ψβ ; τ), we have

[RΩW ](a(t, y))

=

2∂2t + 2 th(t)∂t + 2δ2

y − 6δy +2µ2

ch2(t)− 2 ch(2t)(2πcy)2

− 2 sh(t)(2πcy)τ∨((

1 00 1

))+

2µ th(t)ch(t)

τ∨((

0 1−1 0

))

− 2 ch(t)(2πcy)τ∨((

0 11 0

))− 1

2 ch2(t)τ∨

((0 −11 0

)2 )W (a(t, y)).

Proof. We can prove this by using (4.4) and Lemma 4.5 below. ¤

Lemma 4.5. For any a = a(t, y) ∈ S, the Casimir element Ω can be rewrittenas

Ω =1

2 · ch2(t)Ad(a−1)((H1 −H2)2)

+th(t)ch(t)

Ad(a−1)(H1 −H2)

· κ∗((

0 −11 0

))

+12(H1 + H2)2 − 3(H1 + H2) +

12H2

0 + th(t)H0

+12(th2(t)− 1)κ∗

((0 −11 0

))2

+ y2 Ad(a−1)

4n∗

((ch2(t/2) sh(t)/2sh(t)/2 sh2(t/2)

))2

+ 2n∗

((sh(t) ch(t)ch(t) sh(t)

))2

+ 4n∗

((sh2(t/2) sh(t)/2sh(t)/2 ch2(t/2)

))2

− 4√−1y

Ad(a−1)n∗

((ch2(t/2) sh(t)/2sh(t)/2 sh2(t/2)

))· κ∗

((1 00 0

))

− 2√−1y

Ad(a−1)n∗

((sh(t) ch(t)ch(t) sh(t)

))· κ∗

((0 11 0

))

− 4√−1y

Ad(a−1)n∗

((sh2(t/2) sh(t)/2sh(t)/2 ch2(t/2)

))· κ∗

((0 00 1

)).


Proof. By a direct computation, we have

Ω =12(H1 −H2)2 +

12(H1 + H2)2 − (H1 −H2)− 3(H1 + H2)

+ 2E(1,−1) · E(−1,1) + 4E(2,0) ·

E(2,0) − κ∗

((√−1 00 0

))

+ 2E(1,1) ·

E(1,1) − κ∗

((0

√−1√−1 0

))

+ 4E(0,2) ·

E(0,2) − κ∗

((0 00√−1

)).

It is easy to check that

H1 −H2 = ch(t)−1 Ad(a−1)(H1 −H2) + th(t)κ∗

((0 −11 0

)).

Hence we have

(H1 −H2)2 =

ch(t)−1 Ad(a−1)(H1 −H2) + th(t)κ∗

((0 −11 0

))· (H1 −H2)

= ch(t)−1 Ad(a−1)(H1 −H2)

·

ch(t)−1 Ad(a−1)(H1 −H2) + th(t)κ∗

((0 −11 0

))

+ th(t)

(H1 −H2) · κ∗((

0 −11 0

))+ 2H0

=1

ch(t)2Ad(a−1)((H1 −H2)2)

+2 th(t)ch(t)

Ad(a−1)(H1 −H2)

· κ∗((

0 −11 0

))

+ 2 th(t)H0 + th(t)2κ∗

((0 −11 0

))2

.

The term 2E(1,−1) · E(−1,1) can be rewritten as follows:

1236 T. Moriyama

2E(1,−1) · E(−1,1) =12

H0 − κ∗

((0 −11 0

))·

H0 + κ∗

((0 −11 0

))

=12H2

0 −12κ∗

((0 1−1 0

))2

+ H1 −H2.

The rest of the computation is easy if we notice that Lie(NR) is stable underAd(S). ¤

5. Generalized Whittaker functions belonging to even P1-principalseries representations.

In this section, we suppose that π is equivalent to an even P1-principal seriesrepresentation I(P1; (−1)n ⊗D+

n , ν1) (n ≥ 1, ν1 ∈ C) of G0. By Proposition 2.1,τ = τ(n,n) is a multiplicity one K0-type of π. By using the differential opera-tors introduced in the previous section, we derive a system of partial differentialequations satisfied by the generalized Whittaker function of type (π, χ · ψβ , τ).From these partial differential equations plus a result on the generalized hyper-geometric differential equations (Proposition 9.2), we obtain an explicit formulaof W (g) on a one-parameter subgroup a(0, y) | y > 0 of G0 and prove thatdimC GWmg

G0(π, χ · ψβ) ≤ 1.

5.1. The main results for even P1-principal series representations.We state our main results for even P1-principal series representations.

Theorem 5.1. Suppose that π is equivalent to an even irreducible P1-principal series representation I(P1; (−1)n⊗D+

n , ν1) of G0. We set β =( 0 c/2

c/2 0

)

(c ∈ R×) and fix a quasi-character χ = χµ of T β as in (4.1). Then we have thefollowing assertions:

( i ) dimC GWG0(π, χ · ψβ) ≤ 4.( ii ) dimC GWmg

G0(π, χ · ψβ) ≤ 1.

(iii) Let τ = τ(n,n) be a multiplicity one K0-type of π. For a generalized Whit-taker function W (g) of type (π, χ ·ψβ , τ), we define a C∞-function ϕ(a) onS by

W (a(t, y)) = e−2πcy·sh(t)ynϕ(a(t, y))v0, ∀a(t, y) ∈ S,

where v0 is a fixed non-zero vector in V ∨τ . If W (g) has the moderate growth

property, then we have


ϕ(a(0, y)) = C ×G4,02,4

((πcy)2

∣∣∣∣α1, α2

γ1, γ2, γ3, γ4

),

where C ∈ C is a constant and

α1 =µ + 2

2, α2 =

−µ + 22

,

γ1 =−n + 4 + ν1

4, γ2 =

−n + 4− ν1

4, (5.1)

γ3 =−n + 2 + ν1

4, γ4 =

−n + 2− ν1

4.

Here G4,02,4

(z∣∣ α1, α2

γ1, γ2, γ3, γ4

)stands for the Meijer G-function, whose defini-

tion is recalled in Section 8.

Remark. In view of (3.4), the assertions (i) and (ii) imply thatdimC GWG(π, χ · ψβ) ≤ 4 and dimC GWmg

G (π, χ · ψβ) ≤ 1 when π is an irre-ducible (g,K)-module I(P1; ε⊗D+

n , ν1)[c]. Similar remarks are valid for Theorems6.1 and 7.1 below.

5.2. Proof of Theorem 5.1.The starting point of our proof of Theorem 5.1 is the following:

Proposition 5.2. Let W (g) be a generalized Whittaker function of type(π, χµ · ψβ , τ(n,n)). Then we have

[(P dn · ∇−) (Pup · ∇−)W

](g) = 0, (5.2)

[RΩW ](g) =ν21 + (n− 1)2 − 5

W (g). (5.3)

Proof. The first equation can be easily obtained from Proposition 2.1. Theimage of the Casimir element Ω under the Harish-Chandra isomorphism Z(g0) →U(a)W is given by H2

1 + H22 − 5. This combined with the infinitesimal character

of π given in Subsection 2.2 proves the second equation. ¤

The S-radial parts of the system of differential equations in Proposition 5.2are given as follows:

Proposition 5.3. Let ϕ(a(t, y)) ∈ C∞(S) be as in Theorem 5.1. Then theequations (5.2) and (5.3) are equivalent to the following differential equations

1238 T. Moriyama

− ∂2

t − th(t)∂t + δ2y − δy − µ2

ch2(t)

ϕ(a(t, y)) = 0 (5.4)

and

∂2

t + (−4πcy · ch(t) + th(t))∂t + δ2y + (−4πcy · sh(t) + 2n− 3)δy

+µ2

ch2(t)+

12(n− 2)2 − 1

2ν21

ϕ(a(t, y)) = 0, (5.5)

respectively.

Proof. This can be easily obtained by using Proposition 4.3 (iv), (vi) andProposition 4.4. ¤

We shall derive an ordinary differential equation for ϕ(a(0, y)) from the systemin Proposition 5.3. First we note that, under (5.4), the equation (5.5) can bereplaced by

− 2πcy · ch(t)∂t +

(δy +

n− 2 + ν1

2

)(δy +

n− 2− ν1

2

)

− 2πcy · sh(t)δy

ϕ(a(t, y)) = 0. (5.6)

Since ϕ(a(t, y)) is a real analytic function, we can express the function ϕ(a(t, y))in the form

ϕ(a(t, y)) =∑

j≥0

ϕ〈j〉(y)tj .

Note that ϕ(a(0, y)) = ϕ〈0〉(y). It follows from the equation (5.4) that

−2ϕ〈2〉(y) +(δ2y − δy − µ2

)ϕ〈0〉(y) = 0. (5.7)

From (5.6), we have

−2πcy · ϕ〈1〉(y) +(

δy +n− 2 + ν1

2

)(δy +

n− 2− ν1

2

)ϕ〈0〉(y) = 0 (5.8)

and


− 4πcy · ϕ〈2〉(y) +(

δy +n− 2 + ν1

2

)(δy +

n− 2− ν1

2

)ϕ〈1〉(y)

− 2πcy · δyϕ〈0〉(y) = 0. (5.9)

By eliminating ϕ〈2〉(y) and ϕ〈1〉(y) from (5.7), (5.8), and (5.9), we have

(δy +

n− 4 + ν1

2

)(δy +

n− 4− ν1

2

)(δy +

n− 2 + ν1

2

)(δy +

n− 2− ν1

2

)

− (2πcy)2 ·(

δ2y − µ2

)ϕ〈0〉(y) = 0. (5.10)

We introduce a new variable z = (πcy)2. Then the function φ(z) = ϕ〈0〉(y) satisfiesthe following generalized hypergeometric differential equation

z

2∏

j=1

(δz − αj + 1)−4∏

i=1

(δz − γi)

φ(z) = 0, δz = zd

dz(5.11)

with the parameters αi and γj in the theorem. Since the functions ϕ〈j〉(y) (j >

0) are determined recursively from ϕ〈0〉(y) by (5.6), we have dimC GWG0(π, χ ·ψβ) ≤ 4. Moreover, it follows from Proposition 9.2 plus Lemma 3.3 that thefunction φ(z) coming from an element in GWmg

G0(π, χ · ψβ) is a constant multiple

of G4,02,4

(z

∣∣ α1, α2γ1, γ2, γ3, γ4

). This proves the assertions (ii) and (iii). ¤

Remark. It seems difficult to determine the coefficient function ϕ〈j〉(y) forall j ≥ 0. Following a suggestion of the referee, we express the function ϕ(t, y)

ϕ(t, y) = ch(t)−µ∑

j≥0

ϕ〈j〉(y)(2 sh(t))j

j!.

Then, from the equations (5.4) and (5.5), we have

ϕ〈2j〉(y) =∫

L

(−µ + 12

− s

)

j

∏4i=1 Γ(γi − s)

Γ(α1 − j − s)Γ(α2 − s)(πcy)2s ds

2π√−1

,

ϕ〈2j+1〉(y) =∫

L

(−µ + 22

− s

)

j

∏4i=1 Γ(γi − s)

Γ(α1 − 1/2− j − s)Γ(α2 − 1/2− s)(πcy)2s ds

2π√−1

,

1240 T. Moriyama

for j ≥ 0, where the path L of integration is a loop starting and ending at +∞and encircling all the poles of integrands.

6. Generalized Whittaker functions belonging to odd P1-principalseries representations.

In this section we consider the case where π is equivalent to an odd P1-principal series representation I(P1; (−1)n+1 ⊗D+

n , ν1) (n ≥ 1, ν1 ∈ C) of G0.

6.1. The main results for odd P1-principal series representations.We state our main results for odd P1-principal series representations.

Theorem 6.1. Let π = I(P1; (−1)n+1 ⊗ D+n , ν1) be an odd irreducible P1-

principal series representation of G0. We set β =( 0 c/2

c/2 0

)(c ∈ R×) and fix a

quasi-character χ = χµ of T β as in (4.1). Then we have the following assertions:

( i ) dimC GWG0(π, χ · ψβ) ≤ 4.( ii ) dimC GWmg

G0(π, χ · ψβ) ≤ 1.

(iii) Note that τ = τ(n,n−1) is a multiplicity one K0-type of π. For a generalizedWhittaker function W (g) of type (π, χ · ψβ , τ), we define two C∞-functionsϕk(a) (k = 0, 1) on S by

W (a(t, y)) = e−2πcy·sh(t)ynϕ1(a(t, y))v1 + ϕ0(a(t, y))v0

, ∀a(t, y) ∈ S,

where v0, v1 is a standard basis of V ∨τ . If W (g) has the moderate growth

property, then we have

ϕk(a(0, y)) = (−1)kC ×G4,02,4

((πcy)2

∣∣∣∣α

(k)1 , α

(k)2

γ1, γ2, γ3, γ4

),

where C ∈ C is a constant common to k = 0, 1 and the constants α(k)i and

γj are given by

α(0)1 =

µ + 12

, α(0)2 =

−µ + 22

, α(1)1 =

−µ + 12

, α(1)2 =

µ + 22

,

γ1 =−n + 4 + ν1

4, γ2 =

−n + 4− ν1

4, (6.1)

γ3 =−n + 2 + ν1

4, γ4 =

−n + 2− ν1

4.


6.2. Proof of Theorem 6.1.The starting point of our proof of Theorem 6.1 is the following:

Proposition 6.2. Let W (g) be a generalized Whittaker function of type(π, χµ · ψβ , τ(n,n−1)). Then we have

[(P ev · ∇−)W ](g) = 0, (6.2)

[RΩW ](g) =ν21 + (n− 1)2 − 5

W (g). (6.3)

Proof. This can be proved in the same manner as Proposition 5.2. ¤

The S-radial parts of the system of differential equations in Proposition 6.2are given as follows:

Proposition 6.3. Let ϕk(a(t, y)) ∈ C∞(S) (k = 0, 1) be as in Theorem 6.1.

( i ) The equation (6.2) is equivalent to the system of the equations

(∂t +

12

th(t))

ϕ1(a(t, y))−(

δy +µ

ch(t)− 1

2

)ϕ0(a(t, y)) = 0 (6.4)

and

(δy − µ

ch(t)− 1

2

)ϕ1(a(t, y))−

(∂t +

12

th(t))

ϕ0(a(t, y)) = 0. (6.5)

( ii ) The equation (6.3) is equivalent to the system of the equations

∂2

t + (−4πcy · ch(t) + th(t))∂t + δ2y + (−4πcy · sh(t) + 2n− 3)δy

− 2πcy · sh(t) +µ2

ch2(t)+

12(n− 2)2 − 1

2ν21 −

14(th2(t)− 1)

ϕ1(a(t, y))

+

µth(t)ch(t)

− 2πcy · ch(t)

ϕ0(a(t, y)) = 0 (6.6)

and

1242 T. Moriyama

− µ

th(t)ch(t)

− 2πcy · ch(t)

ϕ1(a(t, y))

+

∂2t + (−4πcy · ch(t) + th(t))∂t + δ2

y

+ (−4πcy · sh(t) + 2n− 3)δy − 2πcy · sh(t) +µ2

ch2(t)

+12(n− 2)2 − 1

2ν21 −

14(th2(t)− 1)

ϕ0(a(t, y)) = 0. (6.7)

Proof. This can be easily proved by using Proposition 4.3 (v) and Propo-sition 4.4. ¤

First we prove the assertion (i) of the theorem. By computing

(∂t +

12

th(t))· (6.4)−

(δy +

µ

ch(t)− 1

2

)· (6.5),

we have

∂2

t + th(t)∂t +14

th2(t) +1

2 ch2(t)−

(δy − 1

2

)2

+µ2

ch2(t)

ϕ1(a(t, y))

+ µ · sh(t)ch2(t)

ϕ0(a(t, y)) = 0. (6.8)

Hence, under (6.4) and (6.5), we can replace the equation (6.6) by

(δy +

n− 2 + ν1

2

)(δy +

n− 2− ν1

2

)− 2πcy · sh(t)

(δy − 1

2

)ϕ1(a(t, y))

− 2πcy · ch(t)(

δy +µ

ch(t)

)ϕ0(a(t, y)) = 0. (6.9)

Similarly the equation (6.7) can be replaced by

− 2πcy · ch(t)(

δy − µ

ch(t)

)ϕ1(a(t, y))

+(

δy +n− 2 + ν1

2

)(δy +

n− 2− ν1

2

)− 2πcy · sh(t)

(δy − 1

2

)

· ϕ0(a(t, y)) = 0. (6.10)


We temporarily put ϕ2(t, y) := δyϕ0(t, y) and ϕ3(t, y) := δyϕ1(t, y). It followsfrom (6.4), (6.5), (6.9), and (6.10) that there exists a set of C∞-functions Ak,l(t, y),Bk,l(t, y) ∈ C∞(Aβ) (0 ≤ k, l ≤ 3) such that

∂tϕk(t, y) =3∑

l=0

Ak,l(t, y)ϕl(t, y) and δyϕk(t, y) =3∑

l=0

Bk,l(t, y)ϕl(t, y),

which proves (i). In order to prove the remaining assertions, we express the func-tions ϕk(a(t, y)) in the form

ϕk(a(t, y)) =∑

j≥0

ϕ〈j〉k (y)tj , k = 0, 1.

From (6.9) and (6.10), we know that

(δy +

n− 2 + ν1

2

)(δy +

n− 2− ν1

2

)ϕ〈0〉1 (y)− 2πcy(δy + µ)ϕ〈0〉0 (y) = 0 (6.11)

and

−2πcy(δy −µ)ϕ〈0〉1 (y)+(

δy +n− 2 + ν1

2

)(δy +

n− 2− ν1

2

)ϕ〈0〉0 (y) = 0, (6.12)

respectively. Eliminating ϕ〈0〉1 (y) or ϕ

〈0〉0 (y) from (6.11) and (6.12), we have

z

2∏

i=1

(δz − α

(k)i + 1

)−4∏

j=1

(δz − γj)

ϕ〈0〉k (z) = 0, k = 0, 1, (6.13)

where z = (πcy)2 and the constants α(k)i and γj are as in the theorem. It follows

from Proposition 9.2 plus Lemma 3.3 that

ϕ〈0〉k (z) = Ck ×G4,0

2,4

(z

∣∣∣∣α

(k)1 , α

(k)2

γ1, γ2, γ3, γ4

)

with some constants Ck. By (6.11), we know that C0 = −C1, which proves theassertion (iii). It can be seen from (6.4) and (6.5) that the functions ϕ

〈j〉k (y)

(k = 0, 1, j > 0) are determined recursively from ϕ〈0〉0 (y) and ϕ

〈0〉1 (y). This proves

(ii). ¤

1244 T. Moriyama

7. Generalized Whittaker functions belonging to (limits of) largediscrete series representations.

In this section we consider the case where π is equivalent to a (limit of) largediscrete series representation.

7.1. The main results for (limits of) large discrete series represen-tations.

Our main results for (limits of) large discrete series representations are asfollows.

Theorem 7.1. Suppose that π is equivalent to a (limit of ) large discreteseries representation D(λ1,λ2) (1 − λ1 ≤ λ2 ≤ 0) of G0. We set β =

( 0 c/2c/2 0

)

(c ∈ R×) and take a quasi-character χ = χµ of T β as in (4.1). Then we have thefollowing assertions:

( i ) dimC GWmgG0

(π, χ · ψβ) ≤ 1.( ii ) Let τ = τ(λ1,λ2) be the minimal K0-type of π. Suppose that W (g) is a gen-

eralized Whittaker function of type (π, χ ·ψβ , τ(λ1,λ2)) with moderate growthproperty. We define C∞-functions ϕk(a) (0 ≤ k ≤ d = λ1 − λ2) on S by

W (a(t, y)) = e−2πcy·sh(t)d∑

k=0

ϕk(a(t, y))vk, ∀a(t, y) ∈ S,

where vk | 0 ≤ k ≤ d is the standard basis of V ∨τ . Then we have

ϕk(a(0, y)) = C × (−1)k ×G4,02,4

((πcy)2

∣∣∣∣α

(k)1 , α

(k)2

γ1, γ2, γ3, γ4

), (0 ≤ k ≤ d),

where C ∈ C is a constant independent of 0 ≤ k ≤ d and the constants α(k)i

and γj are given by

α(k)1 =

−µ + λ1 − k + 22

, α(k)2 =

µ + λ2 + k + 22

,

γ1 =λ1 + λ2 + 4

4, γ2 =

λ1 − λ2 + 44

, (7.1)

γ3 =λ1 + λ2 + 2

4, γ4 =

λ1 − λ2 + 24

.

Remark. It is likely that the estimate dimC GWG0(π, χ · ψβ) ≤ 4 holds


as in the case of P1-principal series representations. But much more computationseems necessary to confirm it. Since our principal interest lies in the subspaceGWmg

G0(π, χ · ψβ), we do not pursue this issue here.

7.2. Differential equations (d ≥ 2).We shall construct a system of partial differential equations satisfied by the

functions ϕk(a). As we shall see later, the case of d = 1 can be reduced to the caseof odd P1-principal series representations. Hence we assume that d ≥ 2. Then,by the location of K0-types of π (Proposition 2.2), we know that a generalizedWhittaker function W (g) of type (π, χµ · ψβ , τ(λ1,λ2)) satisfies the following threeequations:

(P ev · ∇−)W (g) = 0, (P dn · ∇−)W (g) = 0, (P dn · ∇+)W (g) = 0. (7.2)

We rewrite the system (7.2) in terms of the coefficient functions.

Proposition 7.2.

( i ) The equation [P ev · ∇−W ](g) = 0 is equivalent to the system:

− 12(d− k)(d− k − 1) th(t)ϕk+2(a)

+ (k − d)(

δy − µ

ch(t)− λ1 + λ2 + 2

2

)ϕk+1(a)

+ (d− 2k)(

∂t +12

th(t))

ϕk(a) + k

(δy +

µ

ch(t)− λ1 + λ2 + 2

2

)ϕk−1(a)

+12k(k − 1) th(t)ϕk−2(a) = 0, (0 ≤ k ≤ d). (7.3)

( ii ) The equation [P dn · ∇−W ](g) = 0 is equivalent to the system:

12(d− k − 1) th(t)ϕk+2(a) +

(δy − µ

ch(t)− λ1 − 1

)ϕk+1(a)

+ (−2)(

∂t +d + 2

4th(t)

)ϕk(a) +

(∂t +

µ

ch(t)− λ1 − 1

)ϕk−1(a)

+12(k − 1) th(t)ϕk−2(a) = 0, (1 ≤ k ≤ d− 1). (7.4)

(iii) The equation [P dn · ∇+W ](g) = 0 is equivalent to the system:

1246 T. Moriyama

− 12(d− k − 1) th(t)ϕk+2(a) +

(δy − 4πcy · sh(t) +

µ

ch(t)+ λ2 − 1

)ϕk+1(a)

+ 2(

∂t − 4πcy · sh(t) +d + 2

4th(t)

)ϕk(a)

+(

δy − 4πcy · sh(t)− µ

ch(t)+ λ2 − 1

)ϕk−1(a)

− 12(k − 1) th(t)ϕk−2(a) = 0, (1 ≤ k ≤ d− 1). (7.5)

Proof. This can be easily obtained from Proposition 4.3. ¤

The system in the above proposition can be rewritten as follows:

Lemma 7.3. The system of differential equations in Proposition 7.2 is equiv-alent to the following system:

12(k − d)ϕk+1(a) +

(− ∂t +

12(−d + k − 1) th(t)

)ϕk(a)

+(

δy +µ

ch(t)+

k

2− λ1 − 1

)ϕk−1(a) +

12(k − 1) th(t)ϕk−2(a) = 0,

(1 ≤ k ≤ d), (7.6)

− 12(d− k − 1) th(t)ϕk+2(a) +

(− δy +

µ

ch(t)+

λ1 + λ2 + k

2+ 1

)ϕk+1(a)

+(

∂t +12(k + 1) th(t)

)ϕk(a) +

k

2ϕk−1(a) = 0, (0 ≤ k ≤ d− 1), (7.7)

(δy − 2πcy · sh(t)− d + 2

2

)ϕk+1(a)− 4πcy · ch(t)ϕk(a)

+(

δy − 2πcy · sh(t)− d + 22

)ϕk−1(a) = 0, (1 ≤ k ≤ d− 1). (7.8)

Proof. By eliminating the terms involving ϕk+2(a) and ϕk−2(a) from (7.3)and (7.4), we have (7.6) and (7.7), respectively. Moreover, by computing (7.5) +(7.6)− (7.7), we have (7.8). ¤

It is also useful to note that


− 12(d− k − 1) th(t)ϕk+2(a) +

(− δy +

µ

ch(t)+ λ2 + k + 1

)ϕk+1(a)

+12(2k − d) th(t)ϕk(a) +

(δy +

µ

ch(t)− λ1 + k − 1

)ϕk−1(a)

+12(k − 1) th(t)ϕk−2(a) = 0, (1 ≤ k ≤ d− 1), (7.9)

which can be obtained by computing (7.6)k + (7.7)k.

7.3. Proof of Theorem 7.1 (d ≥ 3).In this subsection we prove Theorem 7.1 when d ≥ 3. As in the case of P1-

principal series representations, we express the functions ϕk(a(t, y)) in the form

ϕk(a(t, y)) =∑

j≥0

ϕ〈j〉k (y)tj , 0 ≤ k ≤ d.

From the equations (7.8) and (7.9), we have

(δy − d + 2

2

)ϕ〈0〉k+1(y)− 4πcy · ϕ〈0〉k (y) +

(δy − d + 2

2

)ϕ〈0〉k−1(y) = 0,

(1 ≤ k ≤ d− 1), (7.10)

and

(δy − µ− λ2 − k − 1)ϕ〈0〉k+1(y)− (δy + µ + k − λ1 − 1)ϕ〈0〉k−1(y) = 0,

(1 ≤ k ≤ d− 1), (7.11)

respectively. By eliminating the terms involving ϕ〈0〉k+1(y) and ϕ

〈0〉k−1(y), we have

−2πcy(δy−µ−λ2−k)ϕ〈0〉k (y)+(

δy− λ1 + λ2 + 22

)(δy− d + 2

2

)ϕ〈0〉k−1(y) = 0,

(1 ≤ k ≤ d− 1) (7.12)

and

(δy − λ1 + λ2 + 2

2

)(δy − d + 2

2

)ϕ〈0〉k+1(y)− 2πcy

(δy + µ + k−λ1

)ϕ〈0〉k (y) = 0,

(1 ≤ k ≤ d− 1). (7.13)

1248 T. Moriyama

From these two formulae, we know that each of the functions ϕ〈0〉k (y) (1 ≤ k ≤ d−1)

satisfies

z

2∏

i=1

(δz − α

(k)i + 1

)−4∏

j=1

(δz − γj)

ϕ〈0〉k (z) = 0, z = (πcy)2, (7.14)

where the constants α(k)i and γj are as in Theorem 7.1. Hence it follows from

Proposition 9.2 and Lemma 3.3 that

ϕ〈0〉k (z) = Ck ×G4,0

2,4

(z

∣∣∣∣α

(k)1 , α

(k)2

γ1, γ2, γ3, γ4

), 1 ≤ k ≤ d− 1 (7.15)

with some constants Ck. Moreover by using (7.12)

ϕ〈0〉0 (y) = −C1 ×G4,0

2,4

((πcy)2

∣∣∣∣α

(0)1 , α

(0)2

γ1, γ2, γ3, γ4

)+ φ(y)

with a function φ(y) annihilated by (δy − (λ1 + λ2 + 2)/2)(δy − (d + 2)/2). Sinceϕ〈0〉0 (y) is rapidly decreasing as y → +∞, φ(y) must be identically zero. Similarly

we have

ϕ〈0〉d (y) = −Cd−1 ×G4,0

2,4

((πcy)2

∣∣∣∣α

(d)1 , α

(d)2

γ1, γ2, γ3, γ4

),

and Ck = −Ck+1 (1 ≤ k ≤ d − 2). This proves the assertion (ii). The functionsϕ〈j〉k (y) (j > 0, 0 ≤ k ≤ d) are determined from ϕ

〈0〉k (y) (0 ≤ k ≤ d) via (7.6) and

(7.7). Hence we have dimC GWmgG0

(π, χ · ψβ) ≤ 1.

7.4. Proof of Theorem 7.1 (d = 2).Next we suppose that d = 2, i.e (λ1, λ2) = (2, 0) or (1,−1). By computing

(7.6)k=1 + (7.7)k=1 and (7.6)k=1 − (7.7)k=1, we have

(δy − µ

ch(t)− λ1

)ϕ2(a)−

(δy +

µ

ch(t)− λ1

)ϕ0(a) = 0 (7.16)

and


(δy − µ

ch(t)− λ1 − 1

)ϕ2(a)− 2(∂t + th(t))ϕ1(a)

+(

δy +µ

ch(t)− λ1 − 1

)ϕ0(a) = 0, (7.17)

respectively. We analyze the system of partial differential equations consisting of(7.16), (7.17), (7.6)k=2, (7.7)k=0, and (7.8)k=1. We eliminate ϕ1(a) from (7.6)k=2

by using (7.8)k=1. Then we have

(δy +

µ

ch(t)− λ1 − 1

)(δy − 2πcy · sh(t)− 2)− 4πcy · ch(t)

(∂t +

12

th(t))

ϕ2(a)

+

2πcy · sh(t) +(

δy +µ

ch(t)− λ1 − 1

)(δy − 2πcy · sh(t)− 2)

ϕ0(a) = 0.

(7.18)

Similarly we eliminate ϕ1(a) from (7.7)k=0 and (7.17) by using (7.8)k=1 to get

2πcy · sh(t) +

(δy − µ

ch(t)− λ1 − 1

)(δy − 2πcy · sh(t)− 2)

ϕ2(a)

+(

δy − µ

ch(t)− λ1 − 1

)(δy − 2πcy · sh(t)− 2)− 4πcy · ch(t)

ϕ0(a) = 0

(7.19)

and

2πcy · ch(t)

(δy − µ

ch(t)− λ1

)− (δy − 2πcy · sh(t)− 2)∂t

ϕ2(a)

+

2πcy · ch(t)(

δy +µ

ch(t)− λ1

)− (δy − 2πcy · sh(t)− 2)∂t

ϕ0(a) = 0,

(7.20)

respectively. By setting t = 0 in (7.18), (7.19), and (7.20), we obtain

− 4πcy · ϕ〈1〉2 (y) + (δy + µ− λ1 − 1)(δy − 2)ϕ〈0〉2 (y)

+ (δy + µ− λ1 − 1)(δy − 2)ϕ〈0〉0 (y) = 0, (7.21)

(δy − µ− λ1 − 1)(δy − 2)ϕ〈0〉2 (y)

− 4πcy · ϕ〈1〉0 (y) + (δy − µ− λ1 − 1)(δy − 2)ϕ〈0〉0 (y) = 0, (7.22)

1250 T. Moriyama

and

2πcy(δy − µ− λ1)ϕ〈0〉2 (y)− (δy − 2)ϕ〈1〉2 (y)

+ 2πcy(δy + µ− λ1)ϕ〈0〉0 (y)− (δy − 2)ϕ〈1〉0 (y) = 0, (7.23)

respectively. We eliminate the terms involving ϕ〈1〉2 (y) and ϕ

〈1〉0 (y) in (7.23) by

using (7.21) and (7.22). Then we get

(δy − 2)(δy − 3)(δy − λ1 − 1)− (2πcy)2(δy − µ− λ1)

ϕ〈0〉2 (y)

+(δy − 2)(δy − 3)(δy − λ1 − 1)− (2πcy)2(δy + µ− λ1)

ϕ〈0〉0 (y) = 0. (7.24)

If we set t = 0 in (7.16), then we have

−(δy − µ− λ1)ϕ〈0〉2 (y) + (δy + µ− λ1)ϕ

〈0〉0 (y) = 0. (7.25)

It follows from (7.24) and (7.25) that

z

2∏

i=1

(δz − α

(k)i + 1

)−4∏

j=1

(δz − γj)

ϕ〈0〉k (z) = 0, z = (πcy)2, (7.26)

for k = 0, 2, where the constants α(k)i and γj are as in (7.1). Hence it holds that

ϕ〈0〉k (z) = Ck ×G4,0

2,4

(z

∣∣∣∣α

(k)1 , α

(k)2

γ1, γ2, γ3, γ4

), (7.27)

for k = 0, 2 with some constants C0 and C2. By (7.25), we know that C0 = C2.By setting t = 0 in (7.8)k=1, we have

ϕ〈0〉1 (z) =

12z−1/2(δz − 2)

ϕ〈0〉2 (z) + ϕ

〈0〉0 (z)

= −C0 ×G4,02,4

((πcy)2

∣∣∣∣α

(1)1 , α

(1)2

γ1, γ2, γ3, γ4

),

which proves the assertion (ii) of Theorem 7.1 for the case of d = 2. Finally itis easy to see that ϕ

〈j〉k (y) (j > 0, 0 ≤ k ≤ 2) can be determined recursively from

ϕ〈0〉k (y) (0 ≤ k ≤ 2) by using (7.6)k=1,2 and (7.7)k=0. Hence we have the assertion

(i) for the case of d = 2.


7.5. Proof of Theorem 7.1 (d = 1).Finally we consider the case where d = 1, i.e. π ∼= D(1,0). By comparing the

infinitesimal characters, we know that a generalized Whittaker function W (g) oftype (π, χ · ψβ , τ(1,0)) satisfies the equations (6.2) and (6.3) with ν1 = 0. Hencethe assertions in Theorem 7.1 follow from our computation in Section 6.

Remark. It can be easily checked that D(1,0) is equivalent to a (g0,K0)-submodule of I(P1; 1 ⊗ D+

1 , 0). It is likely that D(1,0) is equivalent to I(P1; 1 ⊗D+

1 , 0) itself. But it is not necessary to verify this for our purpose.

8. Multiplicity free theorems for GSp(2,R).

In this section we derive the multiplicity free results for the case of det(β) < 0from Theorems 5.1, 6.1, and 7.1. Also we reformulate the multiplicity free resultsfor the case of det(β) > 0 in [Mi-1] in the setting of this paper.

8.1. The transition from G to G0.First we clarify the relation between the generalized Whittaker functions on G

and those on G0. For a quasi-character χ of Mβ , we denote the restriction of χ toT β by the same letter. If det(β) > 0, then we have G = RβG0tRβγ0G0 with γ0 =diag(−1,−1, 1, 1) ∈ G. Therefore, the assignment of W (g) to (W (g0),W (γ0g0)),(g0 ∈ G0) gives the following isomorphism

C∞(Rβ\G;χ · ψβ

) ∼= C∞(R1

β\G0;χ · ψβ

)⊕ C∞(R1

β\G0;χ · ψ−β

). (8.1)

Next we suppose that det(β) < 0. Then we have G = RβG0 and Rβ ∩ G0 =±I4R1

β . Hence the restriction map gives the following isomorphism

C∞(Rβ\G;χ · ψβ

)

∼=W ∈ C∞

(R1

β\G0;χ · ψβ

) | W ((−I4)g) = χ(−I4)W (g). (8.2)

Let (π, Hπ) be an admissible smooth representation of G whose central characterωπ : ZR(∼= R×) → C× coincides with the restriction of χ to ZR. Suppose that Hπ

is a direct sum Hπ = Hπ+ ⊕Hπ− of two smooth representations of G0 satisfyingHπ− = π(γ0)Hπ+ . Then it follows from (8.1), (8.2), and χ(−I4) = ωπ(−1) thatthe following isomorphism holds

GWG(π, χ · ψβ) ∼=

GWG0(π+, χ · ψβ)⊕GWG0(π−, χ · ψβ) if det(β) > 0;

GWG0(π+, χ · ψβ) if det(β) < 0.(8.3)

1252 T. Moriyama

We also note that

GWG0(π+, χ · ψ−β) ∼= GWG0(π−, χ · ψβ). (8.4)

8.2. The multiplicity free results for the case of det(β) < 0.Now we can prove the following the multiplicity free results for G =

GSp(2,R).

Theorem 8.1. Let (π;Hπ) be an irreducible (g,K)-module which is equiv-alent to either I(P1;Dn ⊗ ε, ν)[c] (n ≥ 1, ε = ±1, ν ∈ C) or D(λ1,λ2)[c] (1 − λ1 ≤λ2 ≤ 0). Fix a symmetric matrix β ∈ Sym(2)R with det(β) < 0 and take anarbitrary quasi-character χ of Mβ,R. Then we have

( i ) dimC GWmgG (π, χ · ψβ) ≤ 1.

( ii ) If π is equivalent to the representation I(P1;σ, ν)[c] (n ≥ 1, ε = ±1, ν ∈ C),then we have

dimC GWG(π, χ · ψβ) ≤ 4.

Proof. We may assume that χ(zI4) = ωπ(z) (∀z ∈ R×), because otherwisewe have GWG(π, χ · ψβ) = 0. Then in view of (8.3), we know that both of ourassertions are direct consequences of Theorems 5.1, 6.1, and 7.1. ¤

8.3. The multiplicity free results for the case of det(β) > 0.The multiplicity free problem (Problem (A) in the introduction) is discussed in

[Mi-1] for the representations considered in this paper. Although the formulationin [Mi-1] is different from ours, we can paraphrase the results of [Mi-1] as follows:

Theorem 8.2. Let (π, Hπ) be an irreducible (g,K)-module which is equiv-alent to either I(P1;Dn ⊗ ε, ν)[c] (n ≥ 1, ε = ±1, ν ∈ C) or D(λ1,λ2)[c] (1 − λ1 ≤λ2 ≤ 0, λ1 − λ2 ≥ 4). Fix a symmetric matrix β ∈ Sym(2)R with det(β) > 0 andtake an arbitrary quasi-character χ of Mβ,R. Then we have

dimC GWmgG (π, χ · ψβ) ≤ 1.

Proof. As in the proof of Theorem 8.1, we may suppose the compatibilitycondition χ(z) = ωπ(z) (∀z ∈ R×). Then we have

GWmgG (π, χ · ψβ) ∼= GWmg

G0(π+, χ · ψβ)⊕GWmg

G0(π+, χ · ψ−β).

Here π+ is equivalent to I(P1;Dn ⊗ ε, ν) with n ≥ 1, ε = ±1, ν ∈ C or D(λ1,λ2)


with 1− λ1 < λ2 < 0. From now on, we consider the case where π+ is equivalentto an even P1-principal series representation I(P1;Dn ⊗ (−1)n, ν). The othercases can be treated in the same way. In view of (8.4), we may suppose thatβ is positive definite. Further we may suppose that β = I2 without any loss ofgenerality. Recall that there exists a unique vector v0 ∈ π such that π(kA,B)v0 =det(A +

√−1B)nv0 (kA,B ∈ K0) (cf. Proposition 2.1). Take an intertwiningoperator Φ± ∈ GWmg

G0(π+, χ · ψ±β) and set W±

v0(g) := Φ±(v0)(g). By [Mi-1,

p. 261], we can expand each of the functions W±v0

(diag(a1, a2, a−11 , a−2

1 )) as follows:

W±v0

(diag

(a1, a2, a

−11 , a−2

1

))

= (a1a2)n+1e∓π(a21+a2

2)∑

`≥0

p±m0+2`

(a21 + a2

2

)(a21 − a2

2

)m0+2`, ai > 0.

Here m0 is a non-negative integer. It is shown in [Mi-1, (7.8), p. 261] that p±m0(y)

satisfies a second-order ordinary differential equation. Moreover the other coef-ficient functions p±m(y) (m > m0) are determined recursively from p±m0

(y). By asimple calculation, we have

W±(H1−H2)m0v0

(diag(a, a, a−1, a−1)

)

= m0!(2a2)m0 × a2(n+1)e∓2πa2p±m0

(2a2). (8.5)

On the other hand, it is easy to check that

W±(H1−H2)m0v0

(diag(a, a, a−1, a−1);E2,0

)

= ±2π√−1a2W±

(H1−H2)m0v0

(diag(a, a, a−1, a−1)

).

Hence, as in the proof of Lemma 3.3, we conclude that for each N ≥ 0 there existsa constant C > 0 such that

∣∣W±(H1−H2)m0v0

(diag(a, a, a−1, a−1)

)∣∣ ≤ Ca−N , ∀a > 0.

This combined with the asymptotic behavior of the solutions of the ordi-nary differential equations for p+

m0(y) (cf. [Mi-1, p. 261–262]) implies that

dimC GWmgG0

(π+, χ ·ψβ) ≤ 1. Similarly we can conclude that dimC GWmgG0

(π+, χ ·ψ−β) = 0 due to the factor e+2πa2

in the right hand side of (8.5). This proves thetheorem. ¤

1254 T. Moriyama

Remark. We impose the condition λ1 − λ2 ≥ 4, because the computationin [Mi-1, Sections 10–11] is carried out under this assumption. On the other hand,since the computation in [Mi-1, Section 7] for I(P1;Dn ⊗ ε, ν) (n ≥ 2) remainsvalid for n = 1, we do not exclude the case of n = 1 from Theorem 8.2.

9. Rapidly decreasing solutions of a certain generalized hyperge-ometric differential equation.

In this section we prove a result on generalized hypergeometric differentialequations. In order to state it, we recall the definition of the Meijer G-functions(cf. [Er], [Me]).

Definition 9.1. Suppose that m,n, p, and q are integers with q ≥ 1, 0 ≤n ≤ p ≤ q, and 0 ≤ m ≤ q; suppose further that the number z satisfies theinequality 0 < |z| < 1 if q = p, z 6= 0 if q > p, moreover that the numbers ai

(1 ≤ i ≤ p) and bj (1 ≤ j ≤ q) fulfill the condition

ai − bj /∈ Z>0 (1 ≤ i ≤ n; 1 ≤ j ≤ m).

Then the function Gm,np,q (z) is defined as follows

Gm,np,q (z) ≡ Gm,n

p,q

(z

∣∣∣∣a1, a2, . . . , ap

b1, b2, . . . , bq

)

:=∫

L

∏1≤j≤m Γ(bj − s)

∏1≤i≤n Γ(1− ai + s)∏

m+1≤j≤q Γ(1− bj + s)∏

n+1≤i≤p Γ(ai − s)zs ds

2π√−1

, (9.1)

where the path L of integration is a loop starting and ending at +∞ and encirclingall the poles of Γ(bj − s) (1 ≤ j ≤ m) once in the negative direction, but none ofthe poles of Γ(1− ai + s) (1 ≤ i ≤ n).

What we need in this paper is the following proposition, which characterizesthe function G4,0

2,4(z) up to a constant multiple.

Proposition 9.2. Consider the following ordinary differential equation

P (z, δz)φ(z) ≡

z2∏

i=1

(δz − ai + 1)−4∏

j=1

(δz − bj)

φ(z) = 0, ai, bj ∈ C. (9.2)

Let φ(z) be a solution of (9.2) on (0,+∞). Suppose that φ(z) is rapidly decreasingin the sense that for each N > 0 there exists a constant C > 0 such that


|φ(z)| ≤ C × z−N , ∀z ∈ (0,+∞). (9.3)

Then φ(z) is a constant multiple of the G-function G4,02,4

(z

∣∣ a1, a2b1, b2, b3, b4

).

Remark.

( i ) As can be seen from the proof below, under the conditions a1− a2 6∈ Z andai − bj 6∈ Z>0 (∀i, j ∈ Z), Proposition 9.2 is an easy consequence of theasymptotic expansion of G-functions due to Barnes.

( ii ) If we want to extend Proposition 9.2 to the higher order hypergeometricdifferential equations, it seems better to use the general theory of asymptoticexpansions ([Was, Chapters IV–V]).

Proof. We shall prove the proposition by constructing a basis φk(z) | 1 ≤k ≤ 4 for the solution space of the differential equation (9.2). We set

φ3(z) := G4,02,4

(z

∣∣∣∣a1, a2

b1, b2, b3, b4

), φ4(z) := G4,0

2,4

(ze2π

√−1

∣∣∣∣a1, a2

b1, b2, b3, b4

).

Then it is easy to see that φ3(z) and φ4(z) are solutions of (9.2). Moreover it isknown that they satisfy the following estimates due to Barnes ([Ba], see also [Me,p. 131]):

φ3(z) = exp(−2√

z)zϑ(√

π + O(z−1/2)), z → +∞, z ∈ R, (9.4)

φ4(z) = exp(2√

z)(ze2π

√−1)ϑ(√

π + O(z−1/2)), z → +∞, z ∈ R. (9.5)

Here we put ϑ := 1/2(−1/2−∑i=1,2 ai +

∑1≤j≤4 bj). We have to find two other

solutions of (9.2) to make a basis. First we consider the case where a1 − a2 /∈ Z.To each ai (i = 1, 2), we shall attach a solution φi(z) of (9.2) having the property

φi(z) = Ci × zai−1(1 + O(z−1)), z ∈ R, z → +∞ (9.6)

for some constants Ci ∈ C× (i = 1, 2). If a1 − bj /∈ Z>0 for every 1 ≤ j ≤ 4, thenthe function

φ1(z) := G4,12,4

(zeπ

√−1

∣∣∣∣a1, a2

b1, b2, b3, b4

)

is a solution of (9.2) on (0,∞). By shifting the path L to the left and computingthe residue, we know that φ1(z) satisfies (9.6). Next suppose that a1 − bj ∈ Z>0

1256 T. Moriyama

for some 1 ≤ j ≤ 4. We may assume that

a1 − bj ∈ Z>0 (1 ≤ j ≤ m), a1 − bj /∈ Z>0 (m < j ≤ 4);

Re(b1) ≥ Re(b2) ≥ · · · ≥ Re(bm).(9.7)

Put l = a1 − b1 − 1(≥ 0). Then there exists a solution of (9.2) of the form∑lk=0 ckzb1+k with ck 6= 0 for all 0 ≤ k ≤ l, which we denote by φ1(z). Then

(9.6) holds for φ1(z) in this case, too. Interchanging the roles of a1 and a2, wehave a solution φ2(z) of (9.2) satisfying (9.6). It follows from the asymptoticbehavior of φi(z) (1 ≤ i ≤ 4) given above that the set φi(z) | 1 ≤ i ≤ 4 islinearly independent and that any rapidly decreasing solution φ(z) of (9.2) mustbe a constant multiple of φ3(z). Hence the proposition follows when a1 − a2 6∈ Z.

From now on we consider the case where a1 − a2 ∈ Z. Without any loss ofgenerality, we may and do assume that

a1, a2 ∈ Z and a2 ≥ a1. (9.8)

We divide our construction of a basis for the solution space of (9.2) into thefollowing four cases:

Case 1: ai − bj /∈ Z>0 (1 ≤ ∀i ≤ 2, 1 ≤ ∀j ≤ 4);Case 2: bj ∈ [a1, a2 − 1] ∩Z and bk ∈ (−∞, a1 − 1] ∩Z for some 1 ≤ j, k ≤ 4 ;Case 3: bj ∈ [a1, a2 − 1] ∩Z for some 1 ≤ j ≤ 4 and bj /∈ (−∞, a1 − 1] ∩Z for all1 ≤ j ≤ 4;Case 4: bj /∈ [a1, a2 − 1] ∩Z for all 1 ≤ j ≤ 4 and bj ∈ (−∞, a1 − 1] ∩Z for some1 ≤ j ≤ 4.

Case 1: It is easy to see that

φ1(z) = G4,12,4

(zeπ

√−1

∣∣∣∣a1, a2

b1, b2, b3, b4

), φ2(z) = G4,2

2,4

(z

∣∣∣∣a1, a2

b1, b2, b3, b4

)

are solutions of (9.2). If a2 > a1, then φ1(z) and φ2(z) satisfy the estimates (9.6).This proves the proposition in this case. If a1 = a2, then the estimate (9.6) is stillvalid for φ1(z) and

φ2(z) = C2× za1−1(−2γ + log(z))+O(za1−2+ε), z ∈ R, z → +∞, ε > 0, (9.9)

for some constant C2 ∈ C×. Here γ = lims→0(1/s − Γ(s)) is Euler’s constant.Hence the proposition holds for a1 = a2, too.


Case 2: The condition implies that a2 > a1. Then there exist two solutionsφ1(z) and φ2(z) of (9.2) of the form

φ1(z) =a1−1∑

i=bk

cizi, with ca1−1 6= 0, φ2(z) =

a2−1∑

i=bj

c′izi, with c′a2−1 6= 0.

The set φi(z) | 1 ≤ i ≤ 4 of functions on (0,∞) forms a basis for the solutionspace of (9.2). It is then easy to see that the proposition is valid in this case.

Case 3: The condition implies that a2 > a1 and allows us to define a solution

φ1(z) = G4,12,4

(zeπ

√−1

∣∣∣∣a1, a2

b1, b2, b3, b4

)

of (9.2) on (0,∞) satisfying (9.6). Moreover, as in Case 2, there exists a solutionφ2(z) of (9.2) of the form

φ2(z) =a2−1∑

i=bj

c′izi with c′a2−1 6= 0.

The set φi(z) | 1 ≤ i ≤ 4 of functions on (0,∞) forms a basis for the solutionspace of (9.2). Now it is easy to see that the proposition is valid in this case, too.

Case 4: We enumerate bj (1 ≤ j ≤ 4) so that the condition (9.7) holds. Then,as in Case 2, there exists a solution φ1(z) of (9.2) of the form

φ1(z) =a1−1∑

i=b1

cizi with ci 6= 0 (b1 ≤ ∀i ≤ a1 − 1).

We seek for another solution φ2(z) of (9.2). Put φ0(z) := φ2(z) − φ1(z) · log(z).We express the differential operator P (z, δz) as P (z, δz) =

∑4i=0 pi(z)δi with some

polynomials pi(z) (0 ≤ i ≤ 4) in z. We define another differential operator P (z, δz)by

P (z, δz) =4∑

i=1

ipi(z)δi−1z .

Then it is easy to see that

P (z, δz)φ2(z) = P (z, δz)φ0(z) + P (z, δz)φ1(z).

1258 T. Moriyama

First we consider the case of m ≥ 2. Then P (z, δz)φ1(z) belongs to

Cza2−1 ⊕Cza2−2 ⊕ · · · ⊕Czb2+1.

Hence we can find

φ0(z) ∈ Cza2−1 ⊕Cza2−2 ⊕ · · · ⊕Czb2

such that P (z, δz)φ0(z)+P (z, δz)φ1(z) = 0. Hence we have a basis for the solutionof (9.2) and the proposition follows in this case.

Next we suppose that m = 1. In this case, we have

P (z, δz)φ1(z) =a2−1∑

i=b1

c′′i zi for some c′′i ∈ C.

By using m = 1, we have c′′b1 6= 0. Consider the following function

Φ(z) :=∫

L′

Γ(1− a1 + s)Γ(a2 − s)

∏

1≤j≤4

Γ(bj − s)(ze√−1π

)s ds

2π√−1

,

where the path L′ of integration is a loop starting and ending at +∞ and encirclingall the poles of

∏4j=1 Γ(bj − s) once in the negative direction, but none of b1 − k

(k ∈ Z>0). By shifting the path of integration to the left, we know that

Φ(z) =

∏1≤j≤4 Γ(bj − b1 + 1)

Γ(a2 − b1 + 1)Ress=b1−a1 Γ(s)(ze

√−1π)b1−1 + O(zb1−2), (9.10)

when z →∞, z ∈ R. By a simple computation, we have

P (z, δz)Φ(z) =( ∫

L′′−

∫

L′

)Γ(1− a1 + s)∏4

j=1 Γ(bj − s + 1)Γ(a2 − s)

(ze√−1π

)s ds

2π√−1

where the path L′′ is given by L′′ := s + 1 | s ∈ L′. Hence we have

P (z, δz)Φ(z) = αzb1

with


α =

∏4j=1 Γ(bj − b1 + 1)

Γ(a2 − b1)× Ress=b1 Γ(1− a1 + s)× eπ

√−1b1 .

Note that α 6= 0. If a2− 1 = b1, then P (z, δz)− (c′′b1/α)Φ(z) + φ1(z) log(z)

= 0.

If a2 − 1 > b1, then

P (z, δz)− c′′b1

αΦ(z) + φ1(z) log(z)

∈ Cza2−1 ⊕Cza2−2 ⊕ · · · ⊕Czb1+1.

Hence we can find a function

φ(z) ∈ Cza2−1 ⊕Cza2−2 ⊕ · · · ⊕Czb1+1

such that

P (z, δz)φ(z) = P (z, δz)

c′′b1α

Φ(z)− φ1(z) log(z)

.

Summing up, we know that the function

φ2(z) :=

−c′′b1α

Φ(z) + φ1(z) log(z) if a2 − 1 = b1,

φ(z)− c′′b1α

Φ(z) + φ1(z) log(z) if a2 − 1 > b1

is a solution of (9.2). Hence we have a basis for the solution space of (9.2). Nowour assertion follows from (9.4), (9.5), and (9.10). ¤

10. A concluding remark.

In this section, we propose a way of constructing a non-zero element Φ ∈GWmg

G0(π, χ · ψβ), which might be useful for finding the values of the generalized

Whittaker function on the whole group S. This is also an interesting problem fromthe representation theoretic point of view ([G-P]). First we recall the Whittakerfunction on G0 = Sp(2,R). A maximal unipotent subgroup of G0 is given byN0 := N0,R. Any character of N0 can be written as

ψc0,c3 : N0 3 n(x0, x1, x2, x3) 7→ exp(2π√−1(c0x0 + c3x3)

) ∈ C(1)

1260 T. Moriyama

with some c0, c3 ∈ R. We assume that ψc0,c3 is non-degenerate, that is c0c3 6=0. We denote by C∞mg(N0\G0;ψc0,c3) the space of C∞-functions W : G0 → C

satisfying two conditions

• W (ng) = ψc0,c3(n)W (g), ∀(n, g) ∈ N0 ×G0,

• W (g) is of moderate growth.

The group G0 acts on the space C∞mg(N0\G0;ψc0,c3) by right translation. Let(π, Hπ) be an irreducible (g0,K0)-module. Suppose that there exists a non-zerointertwining operator

Ψ : Hπ → C∞mg(N0\G0;ψc0,c3)

for some (c0, c3). We assume that c0 = c3 = 1, which is not an essential restriction.It is known that Ψ is unique up to a constant multiple ([Wal, Theorem 8.8]). Wecall the whole image Ψ(Hπ) the Whittaker model of π. For a Whittaker functionW (g) ∈ Ψ(Hπ) and µ ∈ C, we consider the following integral

ZN (g;µ,W ) :=∫ ∞

0

d×u

∫

R

dx W

1√u

uu

1x 1

w2g

u−µ−1, g ∈ G0.

As we noted in [Is-Mo, p. 5706] (see also [G-P]), this is a variant of Novodvorsky’slocal zeta integral for GSp(2,R). It seems not difficult to show that the integralZN (g;µ,W ) converges absolutely and defines a function in C∞mg(R

1β\G0;χµ · ψβ)

for β =( 0 1/2

1/2 0

)when Re(µ) ¿ 0. Indeed, it is readily seen that our evaluation

[Mo, Proposition 8] of Novodvorsky’s local zeta integrals is compatible with theexplicit formula obtained in Theorem 7.1. On the other hand, in order to get anon-zero element in GWmg

G0(π, χµ · ψβ) for every µ ∈ C, we have to prove the

meromorphic continuability of ZN (g;µ,W ). We hope to discuss it in a futurepaper.

Remark. The assignment W (g) → W (w1g) gives an isomorphism

C∞mg

(R1

β\G0;χµ · ψβ

) ∼= C∞mg

(R1

β\G0;χ−µ · ψβ

).

Hence we have GWmgG0

(π, χµ · ψβ) ∼= GWmgG0

(π, χ−µ · ψβ).


References

[An] A. N. Andrianov, Dirichlet series with Euler product in the theory of Siegel modular

forms of genus two, Trudy Mat. Inst. Steklov., 112 (1971), 73–94.

[An-Ka] A. N. Andrianov and V. Kalinin, Analytic properties of standard zeta-functions of

Siegel modular forms, Mat. Sb. (N.S.), 106 (1978), 323–339.

[Ba] E. W. Barnes, The Asymptotic Expansion of Integral Functions Defined by Gener-

alized Hypergeometric Series, Proc. London Math. Soc., 5 (1907), 59–116.

[Bo] A. Borel, Automorphic forms on SL2(R), Cambridge Tracts in Math., 130, Cam-

bridge University Press, Cambridge, 1997.

[Bu] D. Bump, Automorphic forms and representations, Cambridge University Press,

1997.

[Bu-Fr-Fu] D. Bump, S. Friedberg and M. Furusawa, Explicit formulas for the Waldspurger

and Bessel models, Israel J. Math., 102 (1997), 125–177.

[C-S] C. Casselman and J. A. Shalika, The unramified principal series of p-adic groups,

II, The Whittaker function, Compositio Math., 41 (1980), 207–231.

[Er] A. Erdelyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher transcendental

functions, I, Based on notes left by Harry Bateman, McGraw-Hill Book Company

Inc., New York-Toronto-London, 1953.

[F] M. Furusawa, On L-functions for GSp(4)×GL(2) and their special values, J. Reine

Angew. Math., 438 (1993), 187–218.

[G-P] B. H. Gross and D. Prasad, On irreducible representations of SO2n+1 × SO2m,

Canad. J. Math., 46 (1994), 930–950.

[HC] Harish-Chandra, Discrete series for semisimple Lie groups, II, Acta Math., 166

(1966), 1–111.

[Ha] M. Harris, Occult period invariants and critical values of the degree four L-function

of GSp(4), In: Contributions to Automorphic Forms, Geometry, and Number The-

ory, Johns Hopkins Univ. Press, Baltimore, MD, 2004, pp. 331–354.

[Is] T. Ishii, Siegel-Whittaker functions on Sp(2,R) for principal series representations,

J. Math. Sci. Univ. Tokyo, 9 (2002), 303–346.

[Is-Mo] T. Ishii and T. Moriyama, Spinor L-functions for generic cusp forms on GSp(2)

belonging to principal series representations, Trans. Amer. Math. Soc., 360 (2008),

5683–5709.

[J-S] H. Jacquet and J. A. Shalika, On Euler products and the classification of automor-

phic forms, II, Amer. J. Math., 103 (1981), 777–815.

[Kn] A. W. Knapp, Representation theory of semisimple groups, An overview based on

examples, Princeton Mathematical Series, 36, Princeton University Press, Prince-

ton NJ, 1986.

[K-R-S] S. Kudla, S. Rallis and D. Soudry, On the degree 5 L-function for Sp(2), Invent.

Math., 107 (1992), 483–541.

[Le] F. Lemma, Regulateurs superieurs, periodes et valeurs speciales de la fonction L de

degre 4 de GSp(4), (French, English, French summary) [Higher regulators, periods

and special values of the degree-4 L-function of GSp(4)], C. R. Math. Acad. Sci.

Paris, 346 (2008), 1023–1028.

[Li] J. S. Li, Nonexistence of singular cusp forms, Compositio Math., 83 (1992), 43–51.

[Ma] H. Maass, Siegel’s modular forms and Dirichlet series, Lecture Notes in Math., 216,

Springer-Verlag, 1976.

[Me] C. S. Meijer, On the G-functions I-II, Indag. Math., 8 (1946), 124–134, 213–225.

[Mi-1] T. Miyazaki, The generalized Whittaker functions for Sp(2,R) and the gamma

http://dx.doi.org/10.1112/plms/s2-5.1.59

http://dx.doi.org/10.1007/BF02773797

http://dx.doi.org/10.1515/crll.1993.438.187

http://dx.doi.org/10.1007/BF02392813

http://dx.doi.org/10.1090/S0002-9947-08-04724-7

http://dx.doi.org/10.2307/2374050

http://dx.doi.org/10.1007/BF01231900

1262 T. Moriyama

factor of the Andrianov L-functions, J. Math. Sci. Univ. Tokyo., 7 (2000), 241–295.

[Mi-2] T. Miyazaki, Nilpotent orbits and Whittaker functions for derived functor modules

of Sp(2,R), Canad. J. Math., 54 (2002), 769–794.

[Mo] T. Moriyama, Entireness of the spinor L-functions for certain generic cusp forms

on GSp(2), Amer. J. Math., 126 (2004), 899–920.

[Ni] S. Niwa, On generalized Whittaker functions on Siegel’s upper half space of degree

2, Nagoya Math. J., 121 (1991), 171–184.

[No] M. E. Novodvorsky, Automorphic L-functions for symplectic group GSp(4), In: Au-

tomorphic Forms, Representations and L-Functions, (Eds. A. Borel and W. Clas-

selman), Proc. Sympos. Pure Math., 33, Amer. Math. Soc., Providence RI, 1979,

pp. 87–95.

[No-PS] M. E. Novodvorsky and I. I. Piatetski-Shapiro, Generalized Bessel models for a

symplectic group of rank 2, Math. USSR-Sb., 19 (1973), 243–255.

[PS] I. I. Piatetski-Shapiro, L-functions for GSp4, Olga Taussky-Todd: in memoriam,

Pacific J. Math., Special Issue (1997), 259–275.

[PS-Ra] I. I. Piatetski-Shapiro and S. Rallis, A new way to get an Euler product, J. Reine

Angew. Math., 392 (1988), 110–124.

[Pi-Sch] A. Pitale and R. Schmidt, Bessel models for lowest weight representations of

GSp(4, R), Int. Math. Res. Not., (2009), 1159–1212.

[Pr-TB] D. Prasad and R. Takloo-Bighash, Bessel models for GSp(4), J. Reine Angew.

Math., 655 (2011), 189–243.

[Ro] F. Rodier, Modeles pour les representations du groupe Sp(4, k) ou k est un corps

local, [Models for the representations of Sp(4, k) where k is a local field], Reductive

groups and automorphic forms, I (Paris, 1976/1977), Publ. Math. Univ. Paris VII,

1, Univ. Paris VII, Paris, 1978, pp. 123–145.

[Su] T. Sugano, On holomorphic cusp forms on quaternion unitary groups of degree 2,

J. Fac. Sci. Univ. Tokyo Sect. IA Math., 31 (1985), 521–568.

[TB] R. Takloo-Bighash, Spinor L-functions, theta correspondence, and Bessel coeffi-

cients, with an appendix by Philippe Michel, Forum Math., 19 (2007), 487–554.

[Wal] N. Wallach, Asymptotic expansions of generalized matrix entries of representations

of real reductive groups, In: Lie group representations, I, Lecture Notes in Math.,

1024, Springer-Verlag, 1983, pp. 287–369.

[Was] W. Wasow, Asymptotic expansions for ordinary differential equations, Pure and

Appl. Math., XIV, Interscience Publishers John Wiley & Sons, Inc., New York-

London-Sydney, 1965.

[Y] H. Yamashita, Multiplicity one theorems for generalized Gel’fand-Graev represen-

tations of semisimple Lie groups and Whittaker models for the discrete series, In:

Representations of Lie Groups, Kyoto, Hiroshima, 1986, (eds. K. Okamoto and

T. Oshima), Adv. Stud. Pure Math., 14, Academic Press, Boston, MA, 1988,

pp. 31–121.

Tomonori Moriyama

Department of Mathematics

Graduate School of Science

Osaka University

Machikaneyama-cho 1-1, Toyonaka

Osaka 560-0043, Japan

E-mail: [email protected]

http://dx.doi.org/10.4153/CJM-2002-030-8

http://dx.doi.org/10.1353/ajm.2004.0032

http://dx.doi.org/10.1070/SM1973v019n02ABEH001752

http://dx.doi.org/10.2140/pjm.1997.181.259

http://dx.doi.org/10.1515/crll.1988.392.110

http://dx.doi.org/10.1515/CRELLE.2011.045

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